I'm sorry, folks, the equations weren't present when this text was obtained so it might not make a lot of sense unless you're already familiar with the concepts discussed here.
1. Superstring theory
In this chapter we will discuss some basics of string theory. We will not try to be complete as the subject is much too large to cover here. The main intention here is to give shortly the main ingredients of the theory that we will need and to set some notation.
A string is a one-dimensional extended fundamental object. Just as a point particle moves along a world line in the space-time, the string sweeps out a two dimensional surface while it moves. This surface is called the worldsheet and is denoted. We will only discuss closed strings, so that the string is a circle. The worldsheet s then (at least locally) a cylinder. The world in which the string moves is called the target space M. This target space should be the space-time, but we will see that in general it must be a more general manifold.
The motion of a string is given by an embedding of the worldsheet in the target space. We will generally denote this embedding by X
Missing Equation
This map can be given with respect to coordinates on the target space M. These coordinates then become naturally bosonic scalar fields on the worldsheet. Therefore we see that the string theory is naturally a two-dimensional field theory. Classically this field theory can be defined by an action. A general bosonic action is the nonlinear -model action or Polyakov action
Missing Equation
In this action is the metric on M, an antisymmetric tensor field on M called the Kalb-Ramond field and is the metric on the worldsheet. This action is invariant with respect to coordinate choices on the worldsheet. But furthermore it is also invariant with respect to local scalings of the worldsheet metric
Missing Equation
This symmetry is called Weyl invariance. This symmetry turns out to be crucial for the definition of a quantized theory and allows one to go to a continuum limit (note that the setting we discuss here is just a generalization of first quantization). In fact the action (1.2) is the most general classical Lagrangian which is reparameterization and Weyl invariant.
It is often convenient to make a Wick rotation of the time coordinate on the worldsheet. Using the reparameterization invariance we can then put the worldsheet metric in a standard form. Because of Weyl invariance the (classical) action does not depend on. On a 2-sphere and on the torus we can eliminate this dependence completely, but on a more general worldsheet this can only be done locally. On a two dimensional surface we can use complex coordinates. When the metric is in the standard form with the complex coordinate is given by. For the derivatives we use the conventional shorthands and. The action (1.2) then can be written in the form
Missing Equation
After fixing the metric in the standard form, and thereby choosing complex coordinates, there is still some coordinate invariance left. These are the conformal transformations. In the complex coordinates they are simply the holomorphic and antiholomorphic coordinate transformations. They form an algebra, the conformal algebra or Virasoro algebra. The operators in the field theory described by this action should be in a representation of this algebra. So we essentially have put the string theory in the form of a representation of the conformal algebra. Such a field theory is called a conformal field theory or CFT.
For a consistent quantum theory one has to introduce another term in the action, which is given by
Missing Equation
Here is a scalar field on the target space M called the dilaton. and R(h) is the scalar curvature of the worldsheet. This Lagrangian is not Weyl invariant. But it turns out that the transformation of this part can be used to compensate an anomaly in the Weyl invariance which will turn up after quantization, so this term is really necessary to keep this symmetry. This part of the action has got a nice side effect because it allows to keep track of loops in the string theory. Loops of strings are naturally represented by worldsheets with nontrivial topology. For example one-loop amplitudes can be found by taking for the worldsheet a torus. In general g-loop amplitudes are represented by worldsheets of genus g (that is closed surfaces with g 'holes'). The vacuum expectation value of the dilaton, which gives just the classical contribution from the dilaton, then contributes a purely topological contribution to the dilaton. In fact the integral of the scalar curvature is proportional to the Euler number of the worldsheet
Missing Equation
When we use an Euclidean action we find from this that a g-loop amplitude is multiplied by a factor. Therefore the factor serves as loop-counting parameter in string theory. It can therefore be related to the Planck constant.
After quantizing the theory there is an anomaly in the conformal algebra. This anomaly is determined by the -functions of the theory. The couplings in the -model are the fields G, B and. The -functions are therefore really functional equations.
Ę
where H=dB is the field strength of the B-field. In order to have conformal symmetry in the quantized theory these equations should vanish. Note that to lowest order in these equations give the usual classical field equations of gravity coupled to a scalar field and an antisymmetric tensor field B. For example if the latter two fields vanish we retain the Einstein equation in vacuum. So we should expect to find stringy corrections to these equations of motion if we further expand. We will come back to this remarkable fact later. In the first equations the lowest order -function for the dilaton field is a constant proportional to d-26. This term implies that conformally invariant bosonic -models can only exist on a 26 dimensional target space. This number is therefore called the critical dimension of the bosonic string. This does not seem realistic for a string theory. We will see how this can be solved in a later section.
In the heterotic string model, which we shall discuss later, there is at one loop a correction which modifies the definition of H to
Missing Equation
where denotes the Chern-Simons 3-form of the Lorentz-group (the graviton) and is the Chern-Simons 3-form for the Yang-Mills field present in the heterotic string spectrum. A Chern-Simons 3-form for a gauge field A is defined by
Missing Equation
This correction is related to an anomaly in the effective theory on the target space called the Green-Schwarz anomaly. It has some important consequences as we will see later.
The conformal transformations can be seen as coordinate transformations and are therefore generated by the stress-energy tensor T(z). This tensor is defined as usual by the variation of the action with the (worldsheet) metric. To be precise
Missing Equation
The conformal invariance puts some restrictions on the form of the stress-energy tensor. The Weyl transformations are related to the dilatations. These latter transformations are generated by the trace of the stress-energy tensor. Weyl invariance of the action therefore sets this trace to zero. In complex coordinates this means that
Missing Equation
Hence we keep only two components of the stress-energy tensor. Furthermore these components are really conserved currents, hence they satisfy the conservation law which becomes in the complex notation
Missing Equation
which restricts them to be holomorphic and antiholomorphic functions respectively. We shall from now on denote the two remaining components
Missing Equation
These are precisely the operators which generate the holomorphic and antiholomorphic coordinate transformations which we identified with the conformal transformations.
The modes of the stress-energy tensor are the generators of the Virasoro algebra. They are given by
Missing Equation
This is equivalent to the expansion of T(z) into powers of z (which can also be seen as Fourier modes, if we view z as an exponential)
Missing Equation
The commutation relations of the conformal algebra can be derived using standard methods of quantization. They are given by
Missing Equation
where c commutes with every generator and can therefore be taken a constant. It is called the central charge of the conformal field theory. This central charge is related to the anomaly in the conformal algebra, and thus should vanish for the complete string theory. For the antiholomorphic algebra we can write a similar algebra. Note that the central charge of the two algebras may differ.
The class of string theories that we have discussed until now only has bosonic degrees of freedom. To describe a string theory which also contains fermions in its spectrum, we should include fermionic fields in the worldsheet description. To get a well defined string theory where we can handle potential anomalies, which would spoil an effective space-time description, it turns out that we need supersymmetry on the worldsheet. This means that there is a symmetry present which relates the fermions to the bosons. For a supersymmetric string theory to arise it turns out that the target space should be equipped with a complex structure (see Appendix A). The supersymmetric string theory in -model language is given by the Euclidean action
Missing Equation
where the extra fields are fermionic sections of and R is the curvature tensor determined by the target space metric. Here TM denotes the complexified tangent bundle to M and K is the canonical bundle of. The fermionic fields should be Majorana-Weyl spinors. The equations of motion for the fermions restrict the to be holomorphic and the to be antiholomorphic. This determines the part of the conformal algebra they take part in. The extra terms modify the conformal anomaly equations. One important modification is that the critical dimension is now 10. Therefore a supersymmetric string should live in a 10 dimensional target space.
This model has local N=1 supersymmetry, which means that there is one (local, that is z-dependent) generator for supersymmetry. We will not prove this fact here, but it can be found in the literature, where the Lagrangian is derived in the superfield notation). We will study however string theories which give also supersymmetry in the effective space-time theory. For this to happen we need global N=2 supersymmetry on the worldsheet, in addition to the local N=1 supersymmetry. This means that the one local supersymmetry generator splits globally into two generators. It turns out that this can be achieved by providing the complex target space M with a KŠhler G, in such a way that it becomes Calabi-Yau (see Appendix A for more information). The last condition could in principal be changed if we choose a nontrivial B-field on the target space, but such models are not studied very much in the literature. The N=1 supersymmetry transformations can then be split up to give twice as many supersymmetry transformations.
In addition to the conformal algebra we now have the N=2 supersymmetry transformations. The fact that we have local supersymmetry implies that the algebras mix. Therefore we get a larger algebra which is called the superconformal algebra. For N=2 there are four generators: the stress energy tensor T(z), the supersymmetry generators and a current J(z). As remarked, it follows from the classical equations that the fermions are part of the left-moving algebra and the are part of the right-moving algebra. The supersymmetry generators can be given by properly regularized combinations of the fields
Missing Equation
The -currents J is given by
Missing Equation
which is normalized so that has charge. The current is given by a similar combination, with left-moving replaced by the right-moving.
Because the fields are fermions on the compact worldsheet, they can have nontrivial periodicity conditions. These conditions are not determined, because the Lorentz group on the worldsheet, which is. admits a continuous set of representations. In general we can give the periodicity conditions in terms of a real parameter a by
Missing Equation
For the string theory we will not need the whole range of a. The square of a fermionic field should be bosonic and therefore single valued. The relevant boundary conditions are therefore a=0 which is called the Ramond (R) sector and which is the Neveu-Schwarz (NS) sector. Note that because generators of the supersymmetry transformations are fermionic they satisfy the same boundary conditions.
We can expand, at least locally, the additional operators of the superconformal algebra as in equation(1.15)
Missing Equation
The local N=1 supersymmetry is generated by. The coefficients can again be given by circular integrals, like in equation (1.14). Remark that now the modes of the fermionic operators can be fractional, corresponding to fractional powers of z.
The commutation relations for the modes of these generators are now given by
Missing Equation
Note that the splitting into two components is determined by the charge they have under the current J. It seems that because of the parameter a we have a continuous set of algebras. The algebras for different values of a are however isomorphic. Hence the different forms of the algebra are merely different representations of the superconformal algebra. The isomorphism is given by maps
Missing Equation
where is an arbitrary real number. The last map shows that. so that different values of a are connected. This is called the spectral flow. Note that maps Ramond states to Ramond states and Neveu-Schwarz states to Neveu-Schwarz states. It serves therefore as an automorphism of the sectors. On the other hand maps Ramond states to Neveu-Schwarz states, hence the different sectors are isomorphic. In the space-time this map induces the space-time supersymmetry transformations.
A further enhancement of the supersymmetry algebra can arise if there are two more currents present which together with the current form a level c/6 Kac-Moody algebra generated by the. This means that the modes of these currents satisfy the commutation relations
Missing Equation
Note that the zero modes form an algebra. We will come back more extensively to Kac-Moody algebras later in this chapter. The generators of the supersymmetry can then be split up again into four components determined by their transformation properties under this algebra, giving rise to an N=4 superconformal algebra. These extra supersymmetries can be related to two extra complex structures on the target space which, together with the original complex structure, form a quaternionic structure. This algebra turns up in superconformal theories on hyper-KŠhler manifolds such as the K3 manifold.
Incoming and outgoing states in a CFT correspond to the ends of long tubes attached to the world sheet. Using the conformal invariance of the worldsheet we can however transform these long tubes, such that the worldsheet becomes a `compact' Riemann surface. The states are then represented as punctures on this surface. The in-state is related to a puncture in the point 0, while the out-state is located at. We can define operators which create such a puncture on the worldsheet. So inserting these operators in an expectation value gives the external states. These operators are called local operators. We can view these operators as creating the state out of a vacuum. Often the same notation is used for the state and the operator creating the state. This motivates a notation for the expectation of some operator
Missing Equation
In terms of path integrals we can define states in a slightly better way. We cut out a small disc D around the position of the state. The path integral is an integral over all the values of the fields. We can restrict the field to the worldsheet with the disc cut out. We then have to specify boundary conditions for the fields which take values in a target space M. Let us for definiteness restrict ourselves to purely bosonic fields. As the boundary is simply a circle the boundary condition is a function. This space is called the loop space and is denoted. A state now assigns a weight to every boundary condition, thus it is a function on the loop space. With more external states we have of course as many of these discs cut out. To make contact with the first description of states, we may look at this functional as the path integral over the disc D with the puncture or the operator insertion respectively. This description is also convenient to compare with radial quantization. In radial quantization we use the distance from a puncture as the time variable. The states of radial quantization at a time t can now directly be identified as the state defined as above associated to a disc of radius t. The Hamiltonian operator in radial quantization then relates states associated to discs of different radius, and can therefore be defined as a path integral on the ring between these radii.
The complete set of states is much too general to work with. In general inner products on the Hilbert space of these states can not be defined, and therefore we have no proper definition of expectation values. Furthermore, we want the fluctuations along the worldsheet to vanish as they can be undone by conformal transformations. Also these fluctuations have zero norm or, if they are in a timelike direction of the target space, even negative norm. Therefore we restrict ourselves to a subset of the complete set of states called physical states. Classically we can rule out any fluctuation along the worldsheet by setting the variation with respect to the worldsheet metric equal to zero. We even need to do so as we fixed the metric, thus losing its equation of motion. This means that we should set the stress-energy tensor T to zero. In the quantum theory this could be done by imposing this condition on the states. This can however not be done with all the modes and of the stress-energy tensor, as this is not in accordance with the commutation relations which these operators have to satisfy. Therefore we will only demand the physical state conditions
Missing Equation
and similar relations for the right moving algebra. Here we kept a possibility for a normal ordering constant a, which is necessary in the quantum theory. For the bosonic CFT this is given by a=1. For the supersymmetric CFT it has the value in the NS sector and a=0 in the R sector. A local operator creating a physical state is called a vertex operator.
The Hilbert space of states form a module for the conformal algebra. As the left moving and the right moving algebra decouple completely, we can write the Hilbert space of states as a tensor product of a module for the left moving algebra and one for the right moving algebra
Missing Equation
This means that we break up primitive states as products of a left moving and a right moving piece
Missing Equation
A general state can be written as the sum of these states. From now on we will only consider the left moving states, which form a module for the left moving (super) conformal algebra. The discussion is completely analogous for the right moving states. We can further subdivide this module into submodules for the algebra. Especially we can use states which are eigenstates of the zero modes and of the stress-energy states. The eigenvalues of these operators are called the conformal weights and denoted h and respectively. To build the module of states it is convenient to start with states satisfying
Missing Equation
These states are called primary or highest weight states. We can generate other states, called descendants, by acting on these highest weight states with the operators with n<0. Note that the vertex operators are always primary and have weights (1,1) in the bosonic case and in the fermionic case. This implies that they can be integrated over the (super)worldsheet, thus producing worldsheet Lorentz invariant amplitudes.
Because we use the primary states as the building blocks for the Hilbert space, there are null states, that is states which have inner product zero with any state in the Hilbert space. These are for the bosonic CFT states of the form for n>0 and denoting an arbitrary state. These states decouple from the Hilbert theory and therefore should be set to zero.
In the supersymmetric case we also need fermionic constraints on the primary states. The additional constraints are different in the Ramond sector and the Neveu-Schwarz sector. Also there is an additional quantum number, namely the charge, which we denote q. For both sectors there are constraints on the primary states
Missing Equation
In the Ramond sector we demand in addition
Missing Equation
In the Neveu-Schwarz sector we can set in addition one of the operators to zero. States satisfying the constraint
Missing Equation
are called (left) chiral. The other possibility is called (left) antichiral. We can do the same on the right moving side, obtaining right (anti)chiral primary states.
The primary operators in the superconformal field theory form a natural ring structure. This ring is called the chiral ring. With the above classification we can divide them into subrings according to their chirality. These rings are called the (c,c), (a,c), (c,a) and (a,a) rings respectively where c stands for chiral and a for antichiral, while the first letter refers to the left moving and the second to the right moving sector. Spectral flow relates these rings. It can be shown that the (c,c)-ring is isomorphic to the (a,a)-ring while the (a,c) and the (c,a) are also isomorphic. Therefore there are really two independent rings of primary operators, which we take to be the (c,c)-ring and the (a,c)-ring. We denote these rings and respectively. These rings are the basic building blocks in the construction of the vertex operators of the theory, and therefore these rings determine the spectrum of the string. Remark that there are now additional null states. For example in the (left) chiral sector these are states of the form. These states should be divided out in the chiral rings. After this quotient we obtain the 'physical' chiral ring. We will denote these rings by and likewise for the other rings. We often use the name chiral ring or (c,c)-ring for this quotient as well. Remark that because. this structure is precisely the structure of a cohomology (see Appendix A) with these operators playing the role of d-operator. Also these rings have a natural grading from the charge q of the states. We will see shortly that this charge is positive for chiral states and negative for antichiral states. The operators raise and lower this charge respectively by one unit. we can denote the subrings consisting of operators of (left and right) charge (p,q) by. Then the sign of p and q determine the chirality. To make contact with the general description of cohomology, we see that now these subrings form a cochain, e.g. for the chiral ring. The cohomology now defines the rings which form the ring. and similarly for the other rings. These rings are also often referred to as the ground state rings, as the elements serve as ground states of the theory while the non-primary states can be considered exited states. Later we will see that these rings can naturally be considered the Hilbert spaces for certain small (sub) models related to conformal theories.
In a unitary CFT there are some restrictions on primary states in the Hilbert space. To have Hermitian stress-energy tensor we need a relation between the modes
Missing Equation
and in the supersymmetric case in addition
Missing Equation
From the conformal algebra we derive that the conformal weight of any primary state should be positive from the relation
Missing Equation
Therefore the conformal weight of any state should be positive, as the modes with n<0 can only increase it. From a similar relation with a insertion with n large enough we can also derive that the central charge of a unitary CFT must be positive.
The N=2 superconformal algebra gives even more stringent conditions. These conditions are different for the R and the NS sector, therefore we discuss them separately. In the Ramond sector we derive further constraints from the commutation relations of the zero modes of the supersymmetry generators
Missing Equation
From this we derive that states in the Ramond sector satisfy
Missing Equation
This bound is saturated for the primary states.
In the Neveu-Schwarz sector we derive a bound from the commutation relations of the modes
Missing Equation
This gives a bound. From the other modes, with the switched, we derive a bound. Therefore we obtain the bound in the Neveu-Schwarz sector
Missing Equation
Now we have equality exactly for the (anti) chiral primary states. From the commutation relations we learn that the chiral states must have q positive, while the anti chiral states have q negative. That is
Missing Equation
for (anti) chiral states.
Consider now the ground state. which corresponds to the identity operator of the internal CFT. It has conformal weight h=0 and therefore, as the internal CFT has in general, it cannot be a R state. Therefore it should be a (neutral and primary) NS state. We see from this that this sector has a unique ground state, from which we can build other states. These states are bosonic states from the target space point of view. The R sector has however a degenerate ground state with conformal weight =c/2. It can be inferred that these ground states are in a representation of a (target space) Clifford algebra formed by the zero modes of the fermion fields. Note that as the operators in the R sector have integer mode expansion they have zero modes, contrary to the NS sector. That these zero modes form a Clifford algebra is just the canonical commutation relations for these modes. Therefore the (ground) states in the R sector should form spinors in the target space. This can be checked using more explicit forms of the operators corresponding to these states.
Moduli spaces and string compactifications
In general a moduli space is a space of classes with respect to some equivalence relation. For string theory this relation is the equivalence of conformal field theories. In this full generality this space is of course too complicated, so it is useful to study only a family of conformal field theories and use the parameter space of this family as an approximation of the full moduli space.
A conformal field theory within a certain family can be deformed by adding certain terms to the Lagrangian. These terms are to first order at a given theory given by certain operators belonging to this theory. These operators are called marginal and the corresponding transformations are marginal transformations. To keep conformal invariance, these operators must have conformal dimension (1,1), which is the proper dimension of a Lagrangian as it must be integrated against the (-1,-1) density (this is true for the bosonic case, in the fermionic case we obtain a dimension for these operators). This is precisely the dimension of the vertex operators. This should not be too much of a surprise, as the states of the theory can be viewed as fluctuations in the background of the theory. These operators form a linear space. The parameters of these operators can now serve as local coordinates in the moduli space. In fact this describes the tangent space to the moduli space. In general one has to divide out any symmetry which exist between these operators, such as gauge symmetries. Also the global moduli space is only achieved after dividing out some global equivalence relations that may exist between theories. We will see some examples of this in the following.
If the conformal field theory is given by a -model it can in general be deformed by changing the background fields G and B. This must of course be done in such a way that the theory remains conformal invariant. For the family of supersymmetric theories introduced previously, this means that the metric remains KŠhler to first order in. The B-field can be chosen inside. as exact forms do not contribute to the action. Furthermore the group must be divided out as these forms only change the action by an integer, and therefore do not change the path integral. The full moduli space must also be modded out by all changes which can be undone by diffeomorphisms of M, as these can be undone by coordinate transformations.
As noted before the string theory must live in 26 dimensions if it is described by a bosonic -model and 10 dimensions for a supersymmetric model. These are too many dimensions to describe the 4-dimensional space time we see. A way out of this is to suppose that the target manifold has the form where describes the 4-dimensional Minkowski space and K is a compact manifold to account for the remaining dimensions. This latter manifold is supposed to be very small so that it can not be observed directly.
The physical states are in string theory related to vertex operators which satisfy the physical state conditions as we have seen. In the compactified theory this implies mass shell conditions for these external states. We can build vertex operators from operators in the Minkowski space and in the internal CFT. Equivalently we can build the physical states from tensor products of states in the different CFT's that are involved. The CFT of the flat Minkowski space is quite simple. The zero modes of the relevant states are given by the 4-momentum. There is a momentum operator related to both the left and the right moving CFT. We denote these and respectively. In the Minkowski space these must be equal to the total 4-momentum k due to single valuedness of the fields. The higher modes of these fields and the fermionic fields can give oscillator contributions to the states in the CFT. The conformal weights of the corresponding states in the Minkowski CFT are given by
Missing Equation
where and are the left and right moving oscillator levels respectively. Combining these states with operators in the internal CFT adds the corresponding conformal weights. The physical state conditions on the vertex operators now imply the following mass shell condition
Missing Equation
There is also a more abstract way to get around this phenomenological obstruction. We can construct a string theory by adding the actions of several conformal field theories, which do not have to be conformally invariant by themselves. It is enough if the sum is invariant. The 4-dimensional space-time is then only one of these factors. The other factors do not specifically need to have a geometrical interpretation in general. They can be chosen to be any conformal theory, as long as the central charges - including the charges of the ghost system - add up to zero. The interpretation is then only as giving some internal structure to the strings. In this way truly 4-dimensional strings can be constructed. It is however often very convenient to make some geometrical interpretation. In general we can now try to describe moduli spaces of certain classes of these compactifications or generalizations thereof. Often these can conveniently be described geometrically, which makes them easier to handle. We will see many instances of this later.
The internal conformal field theory describes the most interesting part of the properties of a string model. Therefore we will mainly concentrate on this part, and assume that the space-time sector is set.
There are some classes of manifolds which are of particular interest to be used as compactifications. We have already mentioned that for supersymmetric models the condition on the manifold to be KŠhler already gives an enhancement of supersymmetry, making it possible to do much more calculations. The simplest example of a compact space which is KŠhler is the torus. This also is the simplest manifold one can use as it is locally flat and therefore admits a constant metric and B-field. We will look at this case more careful in a later section.
There is however a drawback to using this kind of manifold for compactification. A supersymmetric string model has N=1 or N=2 supersymmetry on the space-time (this will be discussed in the next section). After compactifying such a string model to 4 dimensions the generators of the 10-dimensional supersymmetry, which are N 32 component spinors break up into 4N 4-dimensional spinors giving rise to N=4 and N=8 supersymmetry depending on the model. This is for a 'realistic' model much too restrictive and therefore one has to break some of these supersymmetries. This can be done by using a more complicated curved compactifying manifold (another way to achieve this is to incorporate instantons in the vacuum; note that in a way we can view a curved manifold as an instanton for the metric field). We will assume that the manifold is complex and the metric is KŠhler, hence being able to use the nice properties of the extended superconformal algebra. In addition to this the metric and the B-field should obey the anomaly equations. It can be shown that also for the supersymmetric string these are still given by the bosonic equations (1.7). This is a consequence of the fact that spinors have a different dimension from bosonic fields, and therefore they contribute as half a momentum which has dimension (remember that is really the only dimensionfull parameter of the string). To lowest order we can solve the equations by setting. and equal to zero. Putting H=dB to zero makes B an element of the second cohomology as exact forms do not contribute to the action on the closed worldsheet. A manifold which has is called called Ricci-flat. A Ricci-flat KŠhler manifold is called a Calabi-Yau manifold. This is then the natural case at which we arrive and which is generally taken for compactification manifold. Note that Calabi-Yau manifolds are always even dimensional as they are complex.
In all even dimensions the torus is a Calabi-Yau manifold when provided with the constant metric. In 2 dimensions this is the only possibility. In 4 dimensions there is also a different choice of Calabi-Yau manifold called a K3 manifold. Geometrically it is a desingularization of the orbifold. where acts by inversion on the torus. Another way to construct a K3 manifold is as the vanishing locus of a homogeneous polynomial of degree 4 in a complex projective space. In 6 dimensions, which is the most interesting case as it compactifies the 10 dimensional space in which the superstring lives to 4 dimensions, there are many more choices. As the Calabi-Yau condition is preserved under taking products one can consider. We will need this manifold later. But there are many more possibilities which can have many different topologies. We will from now on simply denote the latter less trivial cases as Calabi-Yau manifolds.
Until now we only talked about the lowest order equations. In general the KŠhler metric obtains quantum corrections after taking into account the higher order terms. This may change the metric and even the metric will not remain KŠhler in general. The KŠhler class which we will mention later will however remain the same after these corrections. Therefore we are only allowed to use facts that depend on the KŠhler class only. This is however enough. Usually we do not even know the precise form of the KŠhler metric anyway.
In this section we shall meet the most important supersymmetric closed string models in 10 dimensions. As noted in the last section we should 'compactify' in some sense these models on some internal manifold, or more generally replace the 6 internal dimensions by a conformal field theory to obtain a 4-dimensional string. The internal CFT determines the structure of the string model, including its spectrum. But first we will study the possibilities in (flat) 10 dimensional space, as this will serve as a first classification of the possible string theories that can exist. We will concentrate on closed strings, although an open string theory, called type I string, does exist. When we make this restriction it turns out that there are three different string models in 10 dimensions. We will only discuss these models very shortly and concentrate on the massless modes. A more detailed analysis is given in [1].
A superstring theory in 10 dimensions is determined by a superconformal theory with scalars taking values in the flat 10 dimensional space time. The most natural thing to do is to take the (2,2) supersymmetric conformal field theory discussed before. It gives rise to type II string theories. This gives directly a 10 dimensional interpretation from the 10 scalar fields which are present. There is however a small indeterminacy in the fermionic fields. They should be Majorana-Weyl spinors, and therefore we should say which chirality they have. Because the left and right moving sectors decouple we can choose the chirality of the spinor fields for these two sectors separately. There are therefore two cases to be distinguished according to whether they have the same chirality or opposite chirality.
When the chiralities of the spinors are distinct, we obtain the type IIA string. The massless modes are build as product states from the left and the right moving sectors. The target space properties of these massless modes are in general described by the transformation properties with respect to the transverse 'Lorentz' group, which in 10 dimensions is. NS states for the left (right) moving sector are bosonic states from the target space point of view. They can be build from operators and they are in the vector ( ) representation of. The R sector provides the (massless) target space spinors. The vertex operators are in general harder to construct [6] and [5]. The states turn out to be in the spinor ( ) or conjugate spinor ( ) representation, depending on the chirality of the Majorana-Weyl (worldsheet) fermion fields. This gives for the combined left and right moving states the massless spectrum for the type IIA string modes. The content is given by
Missing Equation
where B and F denote the bosonic and fermionic sectors respectively. This string model has N=2 supersymmetry (we will later come back more elaborate on supersymmetry in the target space). The two supersymmetry generators originate fro the left and the right moving sector. Indeed it turns out that the particle content above can be organized into a representation of the N=2 super-PoincarŽ algebra. The NS-NS states give the universal sector of the string, containing the graviton ( ), the antisymmetric B-field ( ) and the dilaton ( ). Furthermore there are R-R states build from the tensor product of the two spinors. These are also bosonic. They organize into a vector and a third rank antisymmetric tensor field (or three-form field). Antisymmetric tensor fields A form a gauge theory in general. Just as for Maxwell theory, which can be considered a special case where the antisymmetric tensor field has rank 1, we have a field strength F=dA, which now is a higher rank antisymmetric tensor. Also there is a gauge symmetry associated to such a field. The supersymmetric partners of all these bosonic modes are in the NS-R and R-NS sector.
When the chirality of the two spinors is the same we obtain a theory is called the type IIB theory. This theory is chiral as a result of the symmetrization. In the same way as for the type IIA string we find for the type IIB string the massless spectrum
Missing Equation
Also this model has N=2 target space supersymmetry. We find again a universal sector, but now the R-R states organize into a scalar. an antisymmetric tensor field (or two-form field) and a rank four self-dual antisymmetric tensor field (four-form field). The self duality here means that the five-form field strength is a self-dual antisymmetric tensor.
As remarked before the superconformal algebra breaks up into two copies corresponding to the left and the right moving sector. The superconformal field theory, being a representation of this algebra, factories accordingly. Because of this factorization we can have different CFT's for the left and the right moving sector. We can even take one sector to be a bosonic CFT, living in 26 dimensions. This theory still has fermions and can therefore be used as a supersymmetry model. In 10 dimensions we can only interpret 10 of the 26 bosons as coordinates on the target space directly. The other 16 (now chiral) bosons form already in 10 dimensions an internal sector. The model which arises in this way is called th heterotic string. A consistent theory exists if we assume that these bosons take values in a 16 dimensional torus. In the next section we will see how this sector can then be described. The torus can be described as where is some 16 dimensional lattice. It follows that in order to have modular invariance for this model this lattice must be even and self dual. Furthermore it is naturally Euclidean. In 16 dimensions there are only two even self-dual Euclidean lattices known. These are the root lattices of the groups and of. The corresponding heterotic strings are accordingly named after these groups. A naive analysis as above for the heterotic string would give a massless spectrum
Missing Equation
Now because we have only one superconformal sector, the target space supersymmetry is N=1. This can also be seen from the particle content above. This contains, apart from the universal NS-NS sector and the corresponding fermions, 16 vectors which are gauge bosons corresponding to the internal 16 chiral bosons, and their superpartners. It turns out however that the 16 bosonic left moving modes can give more massless (bosonic) states. Especially in these two cases we will see that there is a complete nonabelian gauge symmetry present, of which the 16 vectors above are the fields with values in the Cartan algebra. The corresponding gauge group is precisely the group which determines the model. This is a great advantage to the models we saw above, as these have only abelian gauge symmetry - at least perturbatively. This is the reason that the heterotic string is often seen as the most realistic string models.
In this section we will study the simplest but very instructive case where the string moves on a d-dimensional torus. We will only consider the bosonic degrees of freedom. We describe the torus as where is a rank d compactification lattice. Hence the bosonic valued field X is only defined modulo. To define the string action we also need a metric G and an antisymmetric tensor field B. On the torus they will be constant matrices, to assure translation invariance, which will be very important in the following. With this choice the bosonic string action (1.4) taking can be written as
Missing Equation
Using a basis of the lattice we use coordinates on the torus with respect to this lattice:. Note that these coordinates have periodicity. It is often also convenient to use orthonormal coordinates, for which we use indices. for the lattice vectors which therefore satisfy.
The classical equations of motion following from the action (1.47) are
Missing Equation
so that the classical solutions can be written as the sum of a holomorphic and an antiholomorphic part. In the original real coordinates on the worldsheet these are the left and right moving modes respectively. We will often use these terms to denote the different modes. In the conformal algebra these modes are completely decoupled, so they can be quantized separately. We will use radial quantization for this. There is a translational (gauge) symmetry for both the holomorphic and antiholomorphic modes. The corresponding momenta are given by
Missing Equation
These operators are used to construct the stress energy tensor for the theory. These are given by
Missing Equation
and its complex conjugate.
The canonical momentum in radial quantization, generating translations of the complete field. is
Missing Equation
The zero modes of these momenta are
Missing Equation
It follows that the zero mode contribution to the operators and its right moving counterpart, that is the conformal weights, is given by as already alluded to. This means that the energy has a contribution from the zero modes of.
They are restricted by the periodicity of the fields X. First of all, the multivaluedness of X modulo gives the restriction
Missing Equation
This vector is called the winding number for obvious reasons. The translational invariance of X over further restricts the canonical momentum of rigid translations to the dual lattice. We write v for this vector. This is equivalent to the restriction on the momenta in Kaluza-Klein theories. Using this, we can write the holomorphic and antiholomorphic momenta as
Missing Equation
for and. Using the basis of and the dual basis of we can write and with. We then get
Missing Equation
The combined momentum vectors form a rank 2d lattice. Using the Lorentzian inner product given by
Missing Equation
it can easily be shown that this lattice is even and self-dual. In fact, the Lorentzian norm of a lattice vector is given by twice the pairing of and
Missing Equation
and has determinant one.
It is known that all even self-dual lattices of rank (d,d) are related by transformations. On the other hand, all these Lorentzian lattices correspond to toroidal compactifications in the way described above.
For the heterotic string the allowed left and right moving momenta also lie in an even self-dual Lorentzian lattice. But now the number of components of the left moving momenta is not equal to the number of components of the right moving momenta. Therefore the signature of the lattice of momenta is of the form (d+r,d) where d is the dimension of the compactification torus. We denote this lattice. For direct compactification of the standard heterotic string r=16, which is the rank of the gauge group before compactification.
The compactification lattice of the d-torus is denoted. with dual lattice. Furthermore we need an additional even self-dual Euclidean lattice for the extra right moving modes. For the or the heterotic string this can be identified with the corresponding root lattice. Note that the full lattice is again even and self-dual if we use the Lorentzian inner product induced by.
The moduli structure of the compactified heterotic string can be encoded in the geometrical data of the d-torus G and B and in addition the Wilson line moduli A which is a map. These couple the r left moving coordinates to the coordinate fields on the torus. We also use the even self-dual metric C on. The allowed momenta can now given by
Missing Equation
for. and. The Lorentzian norm is given by
Missing Equation
which is again in. using that is even. So the chiral momenta form an even self-dual Lorentzian lattice of signature (d+r,d). All these lattice are now related by transformations, and these transformations act on the heterotic lattice of momenta. The moduli space now has locally the structure of a quotient space. as we will see in a moment. The background fields G, B and A can be shown to parameterize this space. So in general we see that the momentum vector can be written as
Missing Equation
for some linear map
Missing Equation
T Here we denote for any lattice. for the real vector space.he components E, D and A of have a definite geometrical interpretation in this model, as can be seen from the more explicit equation (1.58). There E is the basis of the dual lattice. A are the Wilson moduli and D is related to the metric and the B-field moduli. Sometimes it is however better to view the map as elementary without any reference to any geometry. A direct relation to geometry is even ambiguous in general, as will be clear from the following. The same can be done for. introducing a linear map in addition to. As only the norm of the momentum vectors are important in most cases, the map alone is enough information in view of the equality (1.59).
Because the map is linear we can view it as an element of. Therefore we can take the norm induced by for this map. The resulting element naturally is an element of. where denotes the symmetric tensor product. Using the general formula for the momenta, equation (1.58), we see that it is equal to minus the metric G on. It is however better to see this relation the other way around, and to let G be defined by this relation. So we now define the metric on by
Missing Equation
This has the nice consequence that now the defining equation for the momentum is invariant under general linear transformations of in. including scalings. This is natural as these only relate different equivalent coordinates on the torus. These transformations also kill enough degrees of to get the right number of degrees of freedom to make the identification with the homogeneous space mentioned before. In fact this identification becomes rather simple when we use the parameterization with and. Then the group acts from the right on the lattice and from the left on the moduli field. preserving the inner product on the lattice. Obviously, all relations between different values of the moduli, hence different values of and. must preserve the inner product on the lattice and are therefore elements of this group. There are however also transformations in this group which act as orthogonal transformations in the spaces and. These transformations act trivially on the moduli and therefore must be divided out. This subgroup of trivial transformations can naturally be identified with the maximal subgroup. In this way we obtain the local form of the moduli space
Missing Equation
There is however also a discrete group of trivial transformations which act as global identifications in the moduli space. These are the integral transformations in which preserve the lattice. We will write for this group, as the elements are the matrices with integer coefficients in when written with respect to an integral basis of the lattice. As the lattice, including the Lorentzian norm, is preserved by these transformations it follows that also the spectrum of zero modes given by (1.58) is preserved. This gives already equivalence at the `classical' level. It can be shown that this equivalence is true also in the quantum theory. These transformations must therefore be divided out to get the moduli space. When we look how they act on the geometrical quantities for the torus, we see that they act very nontrivial. To see what can happen we consider the simple case when both B and A are zero. It is easy to see that the transformation which exchanges the vectors v and w (after identification of with ) is in. as it preserves the Lorentzian norm and the lattice. In terms of the r=0 case this is the exchange of the quantum numbers n and m. Then this transformation acts on the remaining modulus G as. This is a highly nontrivial transformation on the geometrical data, as it relates a very large torus to a very small torus. This is a peculiar fact of string theory. It implies that there is an identification in string theory of the large distance effects and the small distance effects. This must really be seen as an identification and not just an equivalence. This fact is partly responsible for the nice convergence property of string theory as it naturally sets a cut off for ultra-violet effects in this way.
The discrete transformations which act on the moduli space of theories are often called duality transformations, as they have some similarity with the identification of electric-magnetic duality. The duality we have just seen is called T-duality or Target space duality, as it is a duality between different geometrical data on the target space. Just as electric-magnetic duality exchanges electric modes with magnetic modes, T-duality also exchanges modes. This is for the particular transformation we just discussed the exchange of winding modes, with the quantum numbers m, and Kaluza-Klein momentum modes, with quantum numbers n. In fact the winding modes can be considered magnetic modes from the worldsheet point of view, as they are really nontrivial topological sectors of worldsheet instantons.
The moduli space of the toroidal compactification is thus given by
Missing Equation
and is called the Narain moduli space.
The whole discussion until now was completely semi-classical, as we only discussed the genus 0 and 1 worldsheet, which corresponds to tree level calculations on the target space. Also we did not look at higher order corrections in. It can however be shown that in general for torus compactifications the dualities in can be corrected perturbatively in and also goes through for higher genus worldsheet. In the latter case, there is a pair of momentum vectors for every pair of intersecting loops on the worldsheet. For a worldsheet of genus g there are g pairs of loops and therefore we need also 2g momentum vectors.
What we discussed here is of course only a tip of T-duality.
We have seen that the moduli space of the toroidal compactified string is given by a homogeneous space. This moduli space is not smooth. There are points in the covering space which are fixed by some elements of the duality group. These points turn up as singularities in the moduli space. We will see later that the effective theory of the string as derived from these compactifications can also have singularities at these points in they moduli space. One could ask what the physical significance of these singularities can be. It turns out that in the CFT these points can have important consequences as there is generally an extra nonabelian gauge symmetry present.
We have seen that physical states correspond to certain local operators in the CFT. These can be split into a left moving and a right moving part in the same way that was done for the states itself. We will again concentrate on the left moving holomorphic part only. For the right movers the story will be completely analogous. In the left moving algebra of the bosonic theory we encountered an important set of local operators, namely the left moving momenta
Missing Equation
The zero modes of these operators generate the rigid translations of the left moving fields. The higher modes are the creation and annihilation operators of the oscillators corresponding to the zero modes of the classical field equation for the left movers. The momentum operators above have conformal weight 1. Therefore they correspond to primary states and we can use them as vertex operators in the theory. Another important set of local operators in the theory are the exponentials
Missing Equation
where k is the momentum. Because the operators P(z) generate the translations it follows that the states generated by these operators are eigenstates of the momentum operators with eigenvalue p, which therefore is the right moving momentum of this state. These operators have conformal dimension. Therefore the operators with could serve as vertex operators. It must however be added that not all these states are in the theory. Indeed, in the last section we have seen that the pair of momentum operators can only have eigenvalues in a certain lattice. Therefore local vertex operators (for the complete left ‡nd right moving CFT) of the form
Missing Equation
can only exist (and do exist) when the momenta are in the lattice. These operators are the pure momentum states. The operators mentioned until now are in fact the only possible states in the bosonic CFT.
What is now the physical interpretation of these operators? A primary operator J(z) for the left moving CFT is really a conserved current of the theory as it satisfies. Therefore it generates a symmetry of the two dimensional theory. Now we can build a vertex operator
Missing Equation
where the coordinate fields are the coordinates on the uncompactified (Minkowski) space. These operators are vectors with respect to the target space Lorentz group. The states related to these operators correspond to gauge bosons of the theory. The fields J, or really the zero modes, are the generators of the corresponding gauge group. We could of course just as well switch the left and right moving part in these vertex operators.
This procedure gives a large number of gauge symmetries in the toroidally compactified string, as any translation operator of the left or right moving CFT can be used for J or the right moving equivalent. Note that these symmetries all commute, therefore the generic gauge group for the compactification of the bosonic string on is. At certain points in the moduli however new operators J or may exist. This happens when we have momentum vectors in the spectrum satisfying
Missing Equation
or the left-right dual vector, as then there are purely (anti)holomorphic momentum states corresponding to a gauge symmetry as explained above. Note that we can also set the second condition to. so that in terms of our parameterization with of the moduli space the condition is
Missing Equation
This condition is exactly the condition for a fixed point of the duality symmetry. This is because the condition is invariant under reflection about. This reflection clearly is a duality transformation as. so that it preserves. The states above have nonzero eigenvalue of the momentum operators. This implies that these operators (or at least the zero modes) do not commute with these operators: they are charged with respect to the abelian gauge group. Therefore the gauge symmetries we get at these special points in the moduli space are nonabelian. So we have found a way to obtain non-abelian gauge symmetries in string theory. The mechanism giving enhanced symmetry we described here is the Frenkel-Kac mechanism. Note that this can only happen for the heterotic string, because in the type II string we have no bosonic CFT.
We can expand, as we did with the other operators we encountered, the currents J(z) in modes
Missing Equation
In general these modes satisfy commutation relations of the form
Missing Equation
where the latter set of commutation relations is required for the currents to be primary. We see that the zero modes of the currents satisfy the commutation relations for the generators of a Lie group if the matrices are the structure constants of this group. Therefore the currents form an infinite dimensional generalization of a Lie algebra. Such an algebra is referred to as a Kac-Moody algebra or a current algebra. The parameter k in the commutation relations of the Kac-Moody algebra is a central charge called the level of the Kac-Moody algebra. For generic gauge groups it should be an integer. A Kac-Moody algebra related to a group G of level k is often denoted. For the simple bosonic model it turns out that the level of the Kac-Moody algebras that arise is always k=1. The value of k restricts in general the possible unitary representations of the algebra. For example when these representations, which are also representations of this group, are restricted by. where l denotes the spin.
As was already remarked the holomorphic momenta give a commuting set of generators (of course modulo the central charge term). Therefore they generate the Cartan algebra (that is the maximal abelian subalgebra) of the Lie algebra. The momentum states that arise at the enhanced symmetry points generate the additional directions in the algebra. Because the state with momentum p has charge p with respect to the Cartan currents these momenta should be identified with the roots of the gauge group. So to find an enhanced gauge symmetry with gauge group G we need a copy of the root lattice of G to exist inside the lattice. which is required to be purely left or right moving. Remember that the uncompactified heterotic string in 10 dimensions has a left moving bosonic sector compactified on a 16-torus. The compactification lattice of this torus is the root lattice of a gauge group, either or. As there is no corresponding right moving sector, the metric is equal to the intersection form which is the self-dual Cartan matrix of the corresponding gauge group. Therefore as already anticipated this model already has a non-abelian gauge group, which is either of the Lie groups given above. The heterotic string is really the only closed string model which admits perturbatively a nonabelian gauge group. This is the reason that the heterotic string is the most popular string model.
For low energies, that is for energies well below the Planck scale, only the massless modes of the string contribute, the exited levels, having a mass at least the Planck mass, giving negligible contributions. Also in this range the typical length scale of the string, which is governed by the parameter and therefore also of the order of the Planck length, is too small to be observed. Therefore it is to be expected that this low-energy behaviour of string theory can be described by a field theory, which is a theory of point particles. The field theory should give a proper description of the low-energy modes. This implies that the correlation functions calculated in this field theory are equal to corresponding amplitudes in string theory.
Note that the field theory should already be quantized to describe the string theory properly, because the higher genus string amplitudes serve as higher loop corrections in the low energy field theory. This is very remarkable if one considers the fact that string theory is really described as a 2-dimensional generalization of a first quantized theory. The Hilbert space for the second quantized string theory, whatever this may be, can however not be described directly in string theory. Multiple string states are not - as in ordinary quantum field theory - defined in the Hilbert space, but are really constructed by taking tensor products of single string Hilbert spaces related to different punctures on the worldsheet. Also space time instantons, which have a clear meaning in field theory as vacuum solutions and generally give non perturbative corrections, can not be realized directly in string theory as they are not described by loops. So what we really have defined up till now is only string theory in perturbation theory. To define the non-perturbative behaviour of string theory we can at the moment only look at the effective field theory. So we will assume that the effective field theory that arises from the (perturbative) string theory does really exist.
The effective Lagrangian can be expanded in powers of. This expansion corresponds to the quantization of the worldsheet theory. This expansion is equivalent to an expansion in powers of momenta and fermions, so that the low energy behaviour consistently is described by the small behaviour. One way to find the Lagrangian is to consider all field theories with the proper spectrum and symmetries to match those of the string theory one considers, and then fix the couplings by comparing the amplitudes. We will see some examples of this procedure.
There is another fruitful method which relies on the following. To have a good vacuum of the theory, all tadpoles (or one-point vertices) should vanish identically. Now such a tadpole is given in string theory by
Missing Equation
where is some vertex operator. Remember that a vertex operator has conformal dimension (1,1), so that this tadpole transforms non-trivially under conformal transformations. Therefore conformal invariance imposes the vanishing of this amplitude. It can be shown that in fact the vanishing of all tadpoles is really equivalent to having conformal invariance. This gives that a choice of vacuum for the theory is in fact equivalent to conformal invariance in the string theory. So the classical equations of motion of the full effective Lagrangian should be the same as the conformal anomaly equations or their generalizations to supersymmetric string theories.
The low energy effective field theory always contains the metric. the antisymmetric tensor field and the dilaton as these correspond to states generated by vertex operators
Missing Equation
which exist in any string theory. The polarization should be taken symmetric and traceless for the graviton, antisymmetric for the B-field and proportional to the identity for the dilaton. These states are referred to as the universal sector. The action for this sector in 10 dimensions for these fields can be found from the anomaly equations (1.7) and is given by
Missing Equation
Note that also the kinetic term of the gravity sector is multiplied by a dilaton factor. This is the way it turns up from the string metric, that is the metric to which the string couples naturally as given by the worldsheet action (1.2). To get the natural metric to obtain the familiar Einstein action the string metric G should rescaled by a dilaton factor. This rescaled metric is referred to as the Einstein metric in the literature.
In the following we will mainly consider the case of N=2 supersymmetric string theories. Therefore the field theory must be some kind of supergravity, which is a supersymmetric extension of (Einstein) gravity. These theories will be the subject of chapter 3.
2. N=2 super Yang-Mills theory
This chapter will be devoted to Yang-Mills theory with extended N=2 supersymmetry. This model was recently solved non-perturbatively in [8] and [9] for the gauge group. These articles meant a revolution in the theory as they gave for the first time an exact solution, thereby describing the mechanism of duality for these nontrivial model. The subject of duality will be a main theme here and in later chapter, so we will be concerned with this subject in more detail. But first of all we will introduce the subject of supersymmetry in more detail, and come to talk about the very important BPS bounds. Lately many introductory notes were produced on these subject.
In globally supersymmetric theories the usual PoincarŽ algebra is extended with additional fermionic symmetries, which exchange bosonic and fermionic fields. The resulting algebra is called the super-PoincarŽ algebra. The possible super-PoincarŽ algebras are characterized by the number of fermionic generators. This number is usually referred to as N. The fermionic generators are then Majorana fermions and ( ). In a unitary theory we take the unitarity condition
Missing Equation
for every generator, so that there are N fermionic charges present. Using the conventions of Wess and Bagger [13], these generators satisfy the following (anti) commutation relations
Missing Equation
Here P is the momentum operator, C is the charge conjugation matrix and is a complex matrix of so called central charges. It commutes with every generator and is antisymmetric in its indices. Usually it is related to the global charges which are present. In any case it should be related to topological quantities and has no dynamical freedom in itself. In the for us most interesting case of N=2 supersymmetry it achieves the form
Missing Equation
as there are only two values for i and j, and hence there is only one (complex) central charge.
The central charge terms play a very important role. They give a lower bound for the masses in the theory. This is really the same condition as the lower bounds for the conformal weights following from the superconformal algebra we derived in section 1.4. From the algebra (2.2) it follows that the mass M of any state in the theory satisfies the bound
Missing Equation
where Z is the largest eigenvalue of the matrix. Remark that in the case of N=2 supersymmetry this is the same as the Z introduced in (2.3). This can be seen by viewing the left hand side of the combined equations (2.2) as a matrix. This matrix must be non negative, as can be seen from the unitary constraint. Therefore the right hand side must also be non negative, which gives the bound. In the N=2 case it is easy to be more explicit. To find the bound we introduce a different basis for the supersymmetry generators
Missing Equation
where is the scaled momentum P/M and is the conjugate charge conjugation, which is simply the inverse. This can really only be done for massive modes. For massless modes one should take the limit in the end. We combine these two components in a vector. The algebra (2.2) now determines the matrix of anticommutation relations
Missing Equation
The left hand side is clearly positive. The right hand side should therefore have non-negative determinant, which is equivalent to the mass bound (2.4). This can easily be seen with respect to a center of mass frame where. which is positive.
The bound we derived above is known as the Bogolmo'nyi-Prasad-Sommerfeld or BPS bound. States which saturate this bound are called BPS saturated states or just BPS states. They play a very important role, because they are the lightest states with given charges and can therefore in general not decay. This of course depends on the precise dependence of the central charges on the other charges. Because a BPS saturated state corresponds to a null vector of the right hand side of the equations (2.2), it must also correspond to a null vector of the left hand side. This implies that a BPS state is invariant under some of the supersymmetry transformations. In general it follows that it preserves precisely one half of the supersymmetry. This really characterizes a BPS state. So we can also see a BPS state as preserving half the supersymmetry transformations. This property can also be used to derive the mass of the state.
An other important point is that if a state saturates the bound classically, it also has to saturate the bound when quantum corrections are included. This can be seen as follows. The generic irreducible representations of the super-PoincarŽ algebra, which do not saturate the BPS bound, have dimension. The representations which do saturate the bound are however lower dimensional: they have dimension. They are therefore called small representations. So if a state classically saturates the bound, then it sits in a small representation. When quantum effects are considered, this small representation can not become larger and hence this state must still be in a small representation. This means that it saturates the BPS bound. This is thus a very powerful tool to determine the masses of certain states, including quantum effects. One must however be careful, because in general the central charges themselves are effected by quantum corrections.
Effective action of N=2 super Yang-Mills
The representations of the super-PoincarŽ algebra are superfields. These are functions not only of the coordinates x of space-time but also of additional anticommuting coordinates. which are spinors. All these coordinates together build what is called superspace. The supersymmetry transformations act on these anticommuting coordinates as some kind of covariant derivative. We will not bother with the details of this, more can be found in e.g. [13]. The components of a superfield with respect to the expansion in the anticommuting coordinates are the usual bosonic and fermionic fields of the underlying field theory. These components form a so called multiplet.
A useful superfield is the chiral superfield. which is a complex superfield with one differential constraint. When we do not include central charges in the algebra it has 16+16 real components (ordinary fields) off-shell, which would imply 16 on-shell degrees of freedom. As mentioned earlier, the minimal multiplet has real on-shell components for N=2. Hence it turns out that the chiral multiplet is reducible. We can reduce the multiplet by an additional constraint on. This gives an irreducible multiplet called a vector multiplet. It has the field content
Missing Equation
where X is a complex scalar, a pair of spinors, a triplet of real scalars (antisymmetric in ab) and a real gauge boson. The vector multiplet is the supersymmetric extension of the gauge field in ordinary field theory. The scalars are auxiliary and can be integrated out. More details on this multiplet can be found in the literature, e.g. [16], [17]. In this discussion the vector is an abelian gauge field. More vector multiplets can be combined into a nonabelian vector multiplet, just as in ordinary field theory.
Another important multiplet is the hypermultiplet. When there is no central charge it has components
Missing Equation
where is a pair of scalar doublets and is a doublet of spinors. When there are central charges present an infinite number of auxiliary fields must be introduced. More details can be found in [17]. These fields play the role of matter fields in the Lagrangian.
The action for supersymmetric theories can be given in terms of integrals over the full superspace, which has coordinates. We will consider N=2 in four dimensions where we have coordinates of each type. Just as the Lagrangian of field theory must be a density to have the right transformation property under coordinate transformations, it follows that the 'superlagrangian' for N=2 must be a chiral superfield. For the N=2 vector multiplet we can find a chiral superfield by taking an arbitrary holomorphic function of the vector superfield. So if denotes the vector multiplet then we define the (ordinary) Lagrangian density by
Missing Equation
where the is an arbitrary holomorphic function. It is called the prepotential. It is this function that we want to determine. When is a nonabelian vector multiplet, or a super Yang-Mills field, the prepotential must be invariant under the gauge group.
Integrating out the anticommuting coordinates projects out the highest component of the superfield. This then gives the Lagrangian. The result can be found in [17]. The bosonic part is given by
Missing Equation
where denotes the self-dual part of the field strength. The indices refer to the different gauge fields. Here and below we often use the shorthand notation
Missing Equation
We will below also use the notation for this matrix of second derivatives. From this it follows that we can identify the matrix of couplings by
Missing Equation
This coupling is complexified as it includes a so-called -angle. This is related to a term in the Lagrangian of the form. Such a term does not affect the classical equations of motion, as it can be written as a total derivative. It does however effects the theory when magnetic monopoles are included [14]. Also the metric on the space where the scalars X take their values, which is determined by the kinetic term for the scalars, turns out to be KŠhler as it can be derived from the KŠhler potential
Missing Equation
At the classical level the prepotential is given by where is some complex constant, which defines the coupling at tree level. In general there will be quantum corrections to the prepotential.
The potential of the classical theory can now be given by
Missing Equation
This potential has flat directions given, up to gauge transformations, by
Missing Equation
where z can be chosen in the complexified Cartan algebra of the gauge group (remember that the vector multiplet and therefore also the scalar X takes values in the Lie algebra of the gauge group). So we see that there is, at least classically, a large vacuum degeneracy. Remark that this vacuum degeneracy persists also after quantization, as the general N=2 scalar potential still contains a commutator. The moduli space of these vacua can be parameterized by the parameter z in the Cartan algebra. Vacuum states corresponding to value of z related by Weyl transformations should be identified as they are related by gauge transformations. The true parameters, which are gauge invariant, is given by the vacuum expectation of the Casimir operators
Missing Equation
At these non-zero vacuum expectation values the gauge fields which do not commute with this VEV will obtain masses by the Higgs mechanism. Generically at large values of the moduli z these masses will be large accordingly. This will break the gauge group generically to an abelian gauge group where r is the rank of the gauge group.
We will from here concentrate completely on the simplest super Yang-Mills theory, which has gauge group. The parameter of the moduli space z then can be chosen to take values in. as the Cartan of is one dimensional. The Weyl group of is and acts on the parameter z by inversion. The only independent Casimir is the quadratic Casimir, so that we have a complex gauge invariant parameter
Missing Equation
on the moduli space. At least for small coupling we should get the asymptotic relation. Semiclassically this relation is valid for all values, but in general quantum corrections will spoil it. At u=0 there is a singularity in the moduli space as the Weyl group acts trivially on this point. There the gauge bosons all become massless, which is the physical interpretation of this singularity. So the classical moduli space is just the complex u-plane punctured at the origin. We will see later that this picture will drastically change when we include the quantum corrections.
In the theory the central charge Z is derived, at least classically, in [18]. It follows that
Missing Equation
where n and m are the electric and magnetic charges respectively (in integer units). This formula is derived from the explicit form of the supersymmetry generators and the derivation makes use of the equations of motions following from the Lagrangian. Therefore this formula will in general have quantum corrections. We will later see how these effect the central charge.
We included a angle in the general Lagrangian. Changing this parameter does not effect the equations of motion. It does however change the physics if we include charged particles especially magnetically charged. There it can be shown that magnetic monopoles obtain an electric charge proportional to [14]. However changing the angle with integer multiples of can not be seen. It can be undone by changing our notion of magnetic charge. Also it changes the Lagrangian by an integer, which does not contribute to any amplitude even in the quantum theory. Remark that this type of duality is completely perturbative in nature. It is given on the parameter by.
There is another type of duality which can not be described in a perturbative manner. This is the famous electric-magnetic duality, which interchanges electric and magnetic charges. This duality is proposed to be an exact symmetry in certain theories in [15]. When there are no sources present, we can easily see this duality in the Lagrangian of the gauge field. Write the kinetic term of the gauge field as
Missing Equation
We can implement the Bianchi identity dF=0, which normally is induced by the definition F=dA, by adding a Lagrange multiplier term
Missing Equation
where is a vector field and is defined through. Integrating out gives the Bianchi identity. We can write the resulting Lagrangian then as
Missing Equation
We can now also view F as a Lagrange multiplier field and integrate it out. This leads to a Lagrangian for the field given by
Missing Equation
So we obtain a Lagrangian of the same form, only now for the dual field strength. This field strength can be related - at least locally - to the old field strength F by
Missing Equation
as the first part in (2.21) must vanish if we integrate out F. This can also be formulated as an interchange of electric and magnetic fields. Therefore this duality is referred to as electric-magnetic duality. Also comparing the dual Lagrangian (2.22) with the original Lagrangian (2.19), we find that the complexified coupling is interchanged with the inverse coupling.
This derivation does not go through directly if we include charges for the gauge field. It can be seen that the dual gauge field couples naturally to magnetically charged fields. As magnetically charged objects, such as magnetic monopoles, are nonlocal with respect to the usual electrically charged field the dual gauge field must couple to the fields in a very nonlocal way. This can be undone by also changing the charged fields in the theory, but then these transformations become nonlocal. This peculiarity can not be avoided.
The dualities we discussed above can naturally be combined to give a duality group. When there are charges present we find that the dualities must preserve the spectrum. This changes the duality group to the discrete group. This group acts on the coupling with fractional linear transformations
Missing Equation
which are generated by the two dualities discussed above. Remark that in a natural way this group acts on the pair of gauge fields in the fundamental representation of the group.
The dualities we discussed here must not be seen as symmetries of the theory. They are just (nonlocal) reparameterizations of the theory. If there are both electric and magnetic charged objects in the spectrum, we see that in the regime where electric charges are weakly coupled the standard representation should be used. On the other hand when the coupling is large the magnetically charged particles are weakly coupled, so that it is more useful to use the dual formulation in terms of.
We will now discuss dualities in the N=2 theory. We have described this theory in terms of the gauge fields and the scalar z. In general it is given in terms of the prepotential. To discuss duality we introduce a dual scalar coordinate. We have seen that the coupling is given by
Missing Equation
Therefore a shift of the angle is induced by a linear shift of
Missing Equation
where the true dualities are given by a integer as discussed before. On the other hand the electric-magnetic duality is induced by
Missing Equation
as can be seen from the form of the coupling given above. So it effectively interchanges the coordinate z and its dual. We find that in general the pair transforms as a vector under the transformations
Missing Equation
It will be convenient to use this pair rather than one of the coordinates together. Just remember that they are dependent.
Also the kinetic term of the scalars is invariant under these duality transformations. This can easily be seen by rewriting this term as
Missing Equation
which is manifestly invariant under the dualities.
The full duality group now acts on the pair in the fundamental representation, as can be seen from the generating dualities discussed above. The prepotential in general changes under duality transformations. This must be done in such a way that the coupling transforms with fractional linear transformations. For example for the electric-magnetic duality we find a transformation which should be equal to the second derivative of the dual prepotential. Therefore we find the definition of z in dual coordinates. from which we derive the relation
Missing Equation
From this we derive the transformation of the prepotential
Missing Equation
Remark that this is really a Legendre transformation. In fact duality transformations are essentially canonical transformations in the phase space of field theory. It can be shown that in general the prepotential changes by quadratic polynomials in the combined coordinates. as we will derive later in a more general setting.
Also the form of the central charge, and therefore of the BPS bound, is slightly different in the general theory. In terms of the coordinates we have introduced it is now given by Missing Equation
for a state of electric and magnetic charge n and m respectively. This formula is manifestly duality invariant, if we also transform the charges (n,m) in the right way under the duality, that is with the conjugate action. This also shows that the shift of must be integral in the presence of magnetically charged states, as only then can the formula for Z be invariant. The BPS bound for the N=2 super Yang-Mills theory then becomes
In unified gauge theories there are in general relations between the various gauge coupling constants for the different factors of the gauge group. These relations are however only valid for momentum scales which are larger than the unification scale. that is for the tree level couplings. For momentum scales less than. there are quantum corrections to the gauge couplings. In general this correction is governed by the renormalization group equation
Missing Equation
Here the beta function is calculated in the broken gauge group factor. At one-loop and at weak couplings the beta function can easily be calculated and is of third order in the coupling
Missing Equation
The coefficient is given by Missing Equation
where is the number of matter multiplets (=hypermultiplets) in the representation and is the Coxeter number of this representation defined by the trace of the squared generator in this representation, i.e.. The representation is the adjoint representation, which is the representation of the gauge vector fields. Using this we can integrate the renormalization group equation. When we also use the relations among the gauge couplings at the scale as boundary conditions, we obtain Missing Equation
for scales. Here contains the remaining terms, which are of order 1 in. This term includes loops over superheavy particles which have masses of the order. It represents therefore the threshold correction to the gauge couplings. It should be remarked that the universal coupling can have threshold corrections, but these do not depend on the particular gauge factor.
Threshold corrections become large when the mass of one of the particles which is integrated out becomes smaller then the mass scale. When the particle becomes massless this even introduces a singularity in the couplings. Let us assume that by some mechanism one multiplet with (electric) charge q has mass m which becomes small. When the threshold corrections become large. The contribution of this multiplet to the threshold corrections were calculated in [19] and are given by Missing Equation
where is the contribution of the multiplet to the -function coefficient. We see a typical logarithmic singularity in this formula. This formula can also be derived directly from the renormalized coupling. When we can use the renormalized coupling (2.37) with the multiplet integrated out When we can use the renormalized coupling (2.37) with the multiplet integrated out. When we should include the multiplet under consideration into our Lagrangian, and not integrate it out. This should give an extra contribution
Missing Equation
to the flow of the coupling. On the other hand, when we do not include the multiplet with mass m into our renormalization and integrate it out we should use the mass scale m as our cut-off. To get the right matching at this mass scale the formula for the threshold correction (2.38) follows.
Weak coupling singularity
The perturbative prepotential can at weak coupling be derived from an anomaly analysis. The classical theory has an R-symmetry, which acts with complex rotations on the scalars. The scalars have charge 2 under this symmetry. Perturbatively this symmetry is broken to a discrete symmetry. A -quotient acts as Weyl symmetries on the scalars, so that there is a symmetry group which acts globally on the moduli space. Remark that this is a true symmetry of theories, and not a reparameterizations, as for the duality group. The symmetry is broken because of an anomaly. It can be shown that the Lagrangian transforms under this anomalous transformation as
Missing Equation
From this we see that the remains, as the instanton number
Missing Equation
is always an integer. These transformations are therefore just -angle transformations. From this anomalous transformation and the form of the Lagrangian in terms of the prepotential we derive that perturbatively the prepotential must have the form Missing Equation
Here is the dynamically generated scale of the theory. So we really have derived a very strong non-renormalization theorem, as the extra term in this expression is really a one-loop expression. This is in accordance with the weak coupling renormalization. For the pure theory (this is the contribution of the two charged gauge bosons of charge ). We can write the one-loop renormalized coupling (2.37) in the form
Missing Equation
Now to find the (non-renormalized) coupling in the Lagrangian, that is the one derived from the prepotential we should take at a certain reference scale (at which we determine the universal scale ). Remark that z scales in the same way as and is really the only quantity that determines the scale in a certain vacuum. Therefore we should take. Inserting this gives the full perturbative answer derived from the anomaly.
From the form of the prepotential derived above we see that the weak coupling regime corresponds to the neighbourhood (remember that the second derivative of the prepotential is related to the inverse coupling). Only there our analysis based on the perturbative anomaly is expected to be valid. We will compactify the moduli space by including the point at infinity in. This point then enters as a singularity. Remark however that this is only a weak singularity as the KŠhler metric on given by only has a logarithmic singularity. This means also that the distance of the point in infinity is of order which is finite, showing that in fact we have to include this point in our moduli space.
Remember that the gauge invariant coordinate on the moduli space is given by
Missing Equation
where X denotes the Higgs field. At weak coupling where quantum corrections are small we thus get the asymptotic relation
Missing Equation
This shows that in this limit z (or rather ) is a good coordinate. The square root reminds of the Weyl group, which acts on the scalar z by inversion.
Because of the perturbative corrections to the prepotential and the corresponding logarithmic singularity we find that the prepotential is multiple valued in this regime. This is best seen by looking at the dual coordinate. From the prepotential we derive the asymptotic formula
Missing Equation
When we go around the singular point at infinity in the moduli space the coordinate u is invariant. For the coordinates however we find a monodromy
Missing Equation
From this formula it is clear that this is a perturbative correction to the classical Weyl reflection. Remark that the monodromy is given by the anomaly for a rotation over. The non perturbative corrections are regular, and can be written as an expansion in. so therefore are single valued. Therefore the monodromies given above are will remain the same also after including non perturbative corrections to the prepotential. In matrix notation this monodromy can be written
Missing Equation
We see that this transformation is a duality transformation. This must be the case for general monodromies in the moduli space as the theory after the monodromy is equivalent to the original theory. And we know already that equivalent formulations are obtained by duality transformations.
We can classify the duality transformations generally in different classes according to their relation to the (locally in the moduli space) perturbative theory. We see in general that the (block) diagonal duality transformations correspond to classical Weyl reflections. The elements a and d (which must be ) are the Weyl reflections themselves. The perturbative monodromies are as we have seen lower triangular, so of the form and correspond to combinations of Weyl reflections and -shifts. The general duality with must be non-perturbative as the coupling is inverted by these transformations so the weak coupling regime is connected to the strong coupling regime.
For a general gauge group this analysis goes through at least qualitatively. We still have the same perturbative dualities related to Weyl symmetries, which now have a block structure.
When we include nonperturbative corrections to the prepotential it can happen that it develops singularities at certain values of the modulus z or its dual. We have seen what this looks like in the weak coupling regime. We even know from this analysis that more singularities must occur in the moduli space. This follows from the fact that the prepotential is a holomorphic function of the moduli, and as the compactified moduli space is identified with (the complex u-plane) and we know that the prepotential is not trivial from a perturbative analysis, it must develop more singularities.
In other cases we must find another origin for these singularities. These can in general be identified with charged particles becoming massless. In general when an electrically charged particle becomes massless we find from (2.38) that the coupling develops a singularity
Missing Equation
where n is the electric charge of the massless particle (in integer units) and m is its mass. As the particle becomes massless it is in a small representation of the supersymmetry algebra and hence must saturate the BPS mass bound. Therefore its mass is given by
Missing Equation
Here we use a local coordinate z, which is zero at the singularity. From the general relation of the coupling to the prepotential we derive asymptotic form of the perturbative prepotential in this regime as
Missing Equation
where we left out the regular single valued terms. This determines the asymptotics of the dual coordinate
Missing Equation
From this we derive a monodromy of the pair
Missing Equation
The same analysis goes through when a magnetic monopole of magnetic charge m becomes massless. Then we should use the dual coordinates and the dual prepotential. In these coordinates we find the same singularity structure
Missing Equation
As in this regime the coordinate is a good coordinate on the moduli space, it must be unaffected by the monodromy. We derive therefore the monodromy in this regime
Missing Equation
This monodromy is related to the electric monodromy by the transformation S. The reason for this should now be clear, as this duality transformations relates the different coordinates in the two regimes. In general the monodromy at a singularity where a dyon of charge (n,m) becomes massless can be found by. where A is a transformation which relates (n,m) to (g,0) (remark that the monodromies and therefore also A act from the right on the charges). Here g must be to assure that it can be related by an transformation. From this we derive that
Missing Equation
This form can also be derived from the fact that the particle that becomes massless itself must be inert under the monodromy.
Remark that the identification of the dyons is not at all unambiguous. Because of the monodromies we only have an identification locally in the moduli space. When we go around the monodromy at infinity we find that for example the monopole (0,1) goes over in a dyon (2,-1). Therefore also the monodromies depend on the precise coordinates that one uses. We should always have this ambiguity in mind.
In general the monodromy at infinity must equal the product of the monodromies at the other singularities. This follows if we deform the path around the point at infinity to a sum of paths around all the other singular points. Therefore the monodromies must satisfy the relation Missing Equation
where the denote the other monodromies. This formula also indicates that there must be several singular points, as is not equal to any single strong coupling monodromy.
What we now have to do to find the global monodromy group is to identify the charges of the particles that become massless. These particles should be stable against decay. From the BPS mass formula we see that generally only particles with charges (n,m) with are stable as otherwise they can decay into particles with proportional charges.
The particles that would come in mind first are the massive gauge fields. We know that classically these fields become massless at the point z=0 where the gauge group should be unbroken. In the N=2 theory this can however not occur, as then the theory would be conformally invariant, giving too much restrictions to get a general N=2 theory (see [8] for more details). This will mean that the point u=0 can not be a singularity.
From this discussion we conclude that we need other singularities in the moduli space. We will assume now that there is a singularity associated to a massless monopole at some point in the moduli space. We will see later that this assumption is really not crucial. We could have taken any particle and arrived at the same solution. Because of the unbroken R-symmetry there should be an other singularity at the point. The assumption of Seiberg and Witten is that these are the only points in he moduli space where particles become massless and therefore are the only singular points that occur apart from the weak coupling singularity. This assumption has not been proved yet, but until now many other low numbers have been excluded, so there is much reason to believe that it is true. As the only scale in the theory is the dynamically generated scale we can relate the location of the singularity to this scale. We will choose parameterizations such that we have precisely. From this assumption we can deduce the monodromy associated to the third singular point from the relation (2.57), as we already know the monodromy at infinity and the monodromy of associated to the monopole
Missing Equation
We obtain
Missing Equation
This is exactly the monodromy associated to a dyon of charge (1,1), indicating that this should be the particle that becomes massless at this point.
The full monodromy group is the group generated by these three monodromies and. This group is the subgroup of consisting of matrices with even off diagonal entries
Missing Equation
To come back to our assumption that a monopole becomes massless, we can see that any stable particle of mass (n,m) with n and m relatively prime can be related by these monodromies to either the charge (0,1) or the charge (1,1). Therefore we see that any choice of stable BPS particle as a candidate to become massless would lead to an equivalent situation.
Let us now summarize what we have learned about the non perturbative moduli space of the N=2 super Yang-Mills theory. We have argued that nowhere in the moduli space the full gauge symmetry is restored. So the gauge group is everywhere equal to. The moduli space has the structure of a with three punctures (the singularities). The monodromy group is as remarked above. Furthermore we know the singularity structure of the prepotential at these singularities. We will see in the next section how these facts are enough to fix the prepotential on the whole moduli space.
We will show how to find the complete solution of the model, including all the instanton corrections. We found that the monodromy in the moduli space of the model is the group. The transformations in this group should be considered exact identifications in the theory. Therefore couplings and where give the same theory (and not just an equivalent theory). So the domain of the complexified coupling constant is not the complex half plane. but the quotient space. This problem is well known in mathematics and is called the Riemann-Hilbert problem. We will now describe an algebro-geometric method to solve this problem. This only one of several possible methods of generating a solution. But this is the one first proposed in [8]. Other methods can be found [20].
We can identify the coupling with the complex structure parameter of a family of elliptic curves (2-tori, see Appendix B for more about elliptic curves). This has the nice property that this parameter is automatically in the half space. Also monodromies are quite conveniently described in this setting. An elliptic curve can conveniently be described as the vanishing locus of a homogeneous polynomial of degree 3 in
Missing Equation
We denote by the corresponding complex structure modulus. The curve will have a singularity at values of u for which there is a solution to the combined equations
Missing Equation
These equations describe a double point, and gives an elliptic curve where one of the cycles is contracted to a point. When we transport the parameter u around this singular point in the moduli space, the parameterization of the curve and also its modulus will in general change. That is, there is a monodromy around this point. Let us denote by the monodromy group of the family of curves at these singular points. So we see that and correspond to the same elliptic curve in the family. For the general family of elliptic curves it is well known that the monodromy group is the complete modular group of elliptic curves. Therefore we must have that. The quotient group describes the group of symmetries in the family, that is it gives equivalent elliptic curves but acts non trivially in the family under consideration. So it could act non trivially on the modular parameter u.
To describe a solution to our super Yang-Mills theory we need to find a family of elliptic curves with exactly the monodromy group around three singular points, which we can identify with. Such a family exists and is in fact unique (which is generally true for a problem of this kind). The solution can be given by the family of elliptic curves
Missing Equation
where we used local coordinates on. so the point at should be added. This curve is a double cover of the parameterized by x. It has branch points at. and x=u. We will choose the branch cuts between the first two points and the last two points.
It is easy to see that this family has singularities at the required points. At these points two of the branch points come together. There is a group of automorphisms of this equation generated by
Missing Equation
This group is a. It should be identified with the quotient group of symmetries. This shows that the monodromy group of the family is. The modulus of the elliptic curve can be defined in general as the quotient of periods of curves Missing Equation
where is the unique (in cohomology) holomorphic one form on the curve given by
Missing Equation
and and form an integral basis of the homology group. Remark that the group acts naturally on this basis, preserving the intersection form. With the elliptic modulus given by (2.65) we see that it acts with fractional linear transformations on this parameter.
From the definition of the coupling in terms of we should have the relation Missing Equation
To match the two ways of writing (2.65) and (2.67) we are led to consider the derivatives dz/du and as periods of curves on. It is natural then to also view the parameters z and as integrals over curves over some one form. As the integral homology group is discrete, the derivative to u does not act on the basis elements. So we must get the one form as a derivative with respect to u of the one form. So we should describe Missing Equation
where is chosen such that
Missing Equation
for some function f(u).
We want that z=0 becomes a singular point only in the classical weak coupling limit, that is when. In this limit the corresponding cycle should vanish. To get this we choose to be the cycle encircling the branch points at. At we have have a singularity at. which corresponds to a vanishing cycle. Therefore we choose for the cycle encircling x=u and. These form a basis of the first homology. The one form can be found from the asymptotic forms of z and. It follows that it can be given by
Missing Equation
As we only need integrals of this form, only the cohomology class is well defined.
Let us now see what happens precisely at the singularities in the moduli space of elliptic curves. At these points there is a vanishing cycle. This means that this cycle has vanishing periods. Geometrically the cycle then contracts to a point. For example at the monopole singularity the cycle contracts to a point as the branch points x=u and coincide. As we have identified strong coupling singularities with massless particles, it seems that we can identify cycles in with particle states. This can also be seen from the BPS mass bound, determined by the central charge (2.33) of a state. Using now the expression of z and as periods we can also give the central charge as a period
Missing Equation
where we have identified the cycle corresponding to the state with charges (n,m). From this formula it now follows directly that the central charge, and therefore the mass, corresponding to a vanishing cycle vanishes. Of course to really have a massless particle at a singular point in the moduli space there should be a particle in the spectrum corresponding to the vanishing cycle.
We did not yet give an explicit solution. This is now straightforward mathematics, and can be solved without too much trouble. A very useful way is to use the so-called Picard-Fuchs equations which can be related directly to the curves as the one above. This is related to the fact that the singularities and monodromies as the one we found naturally turn up in the solutions to certain differential equations. This was already seen and appreciated by Riemann when he proposed and try to solve the Riemann-Hilbert problem ( as it was afterwards called). These methods and generalizations of the discussion can be found in e.g. [21], [22] and [23].
3. Supergravity and special geometry
The effective field theory limit of superstring compactifications to 4 dimensions generally contain gravity. Also they should have certain supersymmetries. These field theories in general are called supergravity theories. We will consider theories which have globally N=2 supersymmetry. These theories are quite restrictive, and much is known about it, and quite general Lagrangians are constructed, cf. [16][17]. We will mention some of the most important properties of these theories. We will be mainly concerned with the couplings of vector multiplets to gravity in these theories. Especially the electric-magnetic dualities in these settings will be discussed in great detail. We will give a very beautiful description of these vector multiplet which arises from these duality considerations and is called special geometry. In the last section we will also discuss the rigid case in a similar formulations and we will see how it arises as a limit when gravity is turned off.
Supergravity is a supersymmetric generalization of ordinary gravity. Therefore it should contain in the action the usual Einstein gravity action
Missing Equation
where R denotes the scalar curvature of the Levi-Civita connection. The N=2 supersymmetric generalization contains many additional fields. In addition to the graviton described by the metric we require now also two spin 3/2 fermionic fields called the gravitini. Together they form a multiplet, the Weyl multiplet. It has the on-shell content
Missing Equation
The canonical momenta of the graviton (the stress-energy tensor) generate local coordinate transformations. In the same way the momenta of the gravitini become the generators of the supersymmetry. In fact the zero modes are related to the generators. Because of the mixing of the local transformations of the gravitini and the supersymmetry transformations (or equivalently the fact that are now local fields) gives that the supergravity theory is really a local version of the rigid supersymmetric theories.
The supergravity theory arising from the Weyl multiplet can couple to fields in other multiplets of N=2 supersymmetry. We have already met the vector multiplet and the hypermultiplet in chapter 2. It turns out that it is even necessary to have such couplings when building a consistent supergravity theory. At least one vector multiplet is needed to break an auxiliary gauge symmetry (see [17]). The scalar and the spinor of this multiplet are auxiliary, but the vector remains physical. It is called the graviphoton.
So what we get is supergravity coupled to n+1 vector multiplets and a number of hypermultiplets. We will only consider the coupling to the vector multiplets and forget about the possible hypermultiplets. What we then get is a local version of N=2 super Yang-Mills theory.
Just as in the rigid case the Lagrangian of local N=2 supersymmetric Yang-Mills theory is completely determined by a holomorphic function F(X) of the scalar fields in the vector superfields. We still call it the prepotential. We denote the n+1 vector superfields, so including the graviphoton, by for. We use indices i for the n ordinary gauge superfields. The Lagrangian for these superfields can be written in the N=2 supersymmetry language
Missing Equation
The where n is the number of vector multiplets will denote the scalars of the theory including a scalar for the graviphoton. This latter scalar represents no additional physical degree of freedom, and hence could be integrated out. In fact, the Yang-Mills theory in this description has an additional local complexified symmetry, the KŠhler-Hodge symmetry, which acts on the scalars with complex scalings. Furthermore, the function F must be a homogeneous function of degree two. The n physical scalar fields parameterize an n-dimensional complex manifold. This can be described by letting the be functions of the local coordinates on this manifold, which we will denote. The functions must be holomorphic in these coordinates and, in view of the non physical additional scalar, are projective sections over. The N=2 supersymmetry requires that the metric on is KŠhler with KŠhler metric given by where the KŠhler potential K is given by Missing Equation
where denotes the derivative of F with respect to. and an analogous notation will be used for multiple derivatives. A manifold with such a metric will be said to be special KŠhler. The curvature of this special KŠhler manifold satisfies the relation Missing Equation
where is a holomorphic tensor of rank three, which is related to the third derivative of the prepotential F(X) by
Missing Equation
As we said before, the whole Lagrangian is encoded in the function F. The kinetic term of the scalars is determined by the KŠhler metric on. The kinetic term of the vector fields, whose field strength we denote by. is given by
Missing Equation
where denotes the self dual part of the field strength and the matrix of (inverse) coupling constants and generalized -angles is given by:
Missing Equation
The symmetric matrix is responsible for anomalous magnetic moments. But maybe more important is the fact that it introduces additional couplings between the fields in different vector multiplets. These couplings are of the nature
Missing Equation
and determine the so-called Yukawa couplings, which are interactions between two fermionic fields and a bosonic field. These couplings will be very important for the determination of the prepotential in certain theories.
Using the homogeneity of the function F(X), it can be shown that the and the satisfy the relation
Missing Equation
Because the matrix should be non degenerate, we see that we just as well could have used the to parameterize. The are really the dual scalars to the. It is again natural to combine the 2n+2 functions together in a vector. The choice of which of them are fundamental (the X's) is still free. To introduce these duality transformations we have to introduce the dual field strengths as Missing Equation
and the complex conjugate expression for. This equation is very similar to the expression for the and the. It is therefore also natural to combine the field strength and its dual into a 2n+2 component vector and the same for the anti-selfdual part. This vector is of course related by N=2 supersymmetry to the corresponding vector we introduced for the scalars. The Bianchi identities and equations of motion for the abelian gauge fields can be written as
Missing Equation
These equations together are invariant under the duality transformations Missing Equation
where S is a real constant matrix. Transformation of the anti-selfdual parts follow by complex conjugation. However to ensure that the relation (3.11) remains valid with a symmetric tensor. the transformation (3.14) must be symplectic, that is: the matrix S must be an matrix. Because of N=2 supersymmetry, the scalars also transform under this duality transformation
Missing Equation
Remark that the Lagrangian is not invariant, only the combined Bianchi identities and equations of motion are. This is also reflected through the transformation of the matrix. which transforms with a fractional linear transformation: if we write the matrix S as
Missing Equation
then transforms as:
Missing Equation
So we see that the fields we used are not canonically defined. It seems that they can be redefined with the help of the symplectic transformations introduced above. We are even forced to take this point of view because of a some further generic properties of the theory. The space generically has singularities, that is: points where the theory as described above is not well defined. When we transport the fields around such a singularity, the theory should of course not change. On the other hand, the fields have a monodromy around such a point. This monodromy is, as it should be to get an equivalent theory, an duality transformation. Therefore these transformations should not be looked upon as equivalences of different theories but merely as a reformulation of the same theory. This is of course only true in the classical theory. It turns out that quantum effects, especially the non perturbative aspects, break the duality group down to.
This can also be seen from the spectrum of the theory. Of course the true dualities should preserve this spectrum. Remember that from the super-PoincarŽ algebra we know that the spectrum of BPS states is restricted by the central charges. For an arbitrary (solitonic) state with electric and magnetic charges with respect to the gauge fields given by the complex central charge is given in supergravity by Missing Equation
Remark that the normalization with the KŠhler potential is needed to make the formula invariant under the KŠhler-Hodge symmetry. The formula for the electrically charged states is really quite straight forward, and is a direct generalization of the rigid case. The magnetic generalization is then determined by duality. The formula for the central charge is now manifestly duality invariant if we also transform the charges. Charge quantization restricts the charges (n,m) to integers, therefore the symplectic duality transformations are restricted to the integral group.
This description is further analyzed in [24] and [25]. We will however take a slightly different look at it and consider a more mathematical formulation of special geometry. This has the advantage that it is fully coordinate invariant and makes directly clear which are the essential ingredients of the structure.
The natural mathematical way to look at the situation described above is to say that the scalars are not just a set of scalar functions, but are sections of some 2n+2-dimensional vector bundle over. The monodromies which might arise are then just global properties of this bundle. The duality transformations suggest that should be an associated bundle with structure group. Furthermore, this bundle must have a holomorphic structure, because the scalars are holomorphic functions. The section described by the scalars is a holomorphic section of and spans a holomorphic line subbundle which we denote by. This abstract way to look at special geometry will be the subject for the next section.
The description we will give of special geometry was first formulated in [26]. A more correct formulation can be found in [27], where the formulation was generalized to other settings. To give a coordinate invariant definition of special geometry, we start with the holomorphic symplectic bundle over and the line bundle. Hence we define special geometry as a KŠhler manifold provided with the following additional structure:
1. A holomorphic bundle over with a compatible Hermitian metric.
2. A holomorphic line subbundle of such that the first Chern class of is equal to the KŠhler form of :
Missing Equation
The second part implies that is restricted KŠhler. One usually defines the holomorphic subbundle by choosing a holomorphic section in every coordinate patch. Different choices of this section then are related by scaling with a holomorphic function. The scalar functions (X,F) from the last section are just the coordinates of in. Because is a holomorphic vector bundle with a real structure group there is a natural flat connection. This connection can equivalently be defined as the connection which preserves the Hermitian metric. To get the results of special geometry we must have one further restriction on the line bundle :
3. The first and second derivatives of sections of the line bundle must be orthonormal to this bundle, i.e. the holomorphic section satisfies
Missing Equation
The second statement in our definition is equivalent to saying that the KŠhler potential K of the KŠhler metric on can be found from Missing Equation
for the choice of section.
Although is the natural connection on the symplectic bundle, it is not the natural connection for studying the variation of the moduli. The (classical) structure of the theory is encoded in the section as we have seen in the last section. So the variation of this structure is in some way given by the derivatives. These derivatives contain however a component in the direction of. as it follows from Missing Equation
That we do not have the most natural connection yet also follows from the fact that we can scale our choice of section with a holomorphic function
Missing Equation
for any holomorphic function f(z). This is just the KŠhler-Hodge structure mentioned before. The KŠhler potential transforms as
Missing Equation
which does not change the KŠhler metric or any other physical quantity. So to get the natural objects to study the variation of moduli, we must abandon in some way this judicious choice. We can do this by dividing out the line bundle. which is trivialized by. So we look at the bundle. Now there is a natural section of the line subbundle because this bundle is now trivial. The embedding of in gives a natural connection on this bundle induced by. This connection is given on any section of (so that f(z) is a holomorphic function) by
Missing Equation
which can easily be derived using (3.22). This also determines a connection on the dual bundle. Together with the natural connection on this determines a connection D on the bundle. It is given on sections of by
Missing Equation
After choosing a section we can identify the bundle with. Hence we can view D locally as a connection on in this case.
The bundle can be decomposed into several subbundles
Missing Equation
In this decomposition and. The subbundle is defined as the part orthonormal to in the image of the connection. We want to define the theory. The (orthonormal) holomorphic variations of. given by the vectors. should then parameterize the variations in the moduli space. Therefore these components must be independent, as they are related to independent variations in the moduli space. Hence this subbundle is spanned by the image of the holomorphic tangent bundle of the -valued one-form. The bundle is defined as the dual bundle in of. With this definition we can identify with and in the same way. Hence we have effectively the decomposition
Missing Equation
This decomposition shows why it is often convenient to divide out the bundle. because then the holomorphic tangent bundle T is naturally embedded in this bundle.
To make contact with the description of special geometry, we have to define the quantities that were introduced there in terms of the bundles and. We already know how to find the KŠhler potential. Because the symplectic bundle is flat we can find a symplectic real basis of local flat sections ( ) which satisfy
Missing Equation
We can expand with respect to this basis
Missing Equation
Remark that now the coordinates and are holomorphic functions on. Often we choose the basis in such a way that the provide good projective coordinates on. This can always be done as follows from the fact that the first derivatives of span n independent directions of. These are then called projective special coordinates. The can then be viewed as functions of homogeneous degree 1 of the. Also from the general definition (3.21) of the KŠhler potential the formula (3.4) can now easily be derived in special coordinates, indicating that we made the right identification. This formula is even valid when the are not good coordinates.
The relation can be written with respect to these coordinates as
Missing Equation
If the are good projective coordinates on it follows from this equality that we can write the dual coordinates as
Missing Equation
where
Missing Equation
which is a holomorphic function of homogeneous degree two of the scalars. Hence we have recovered the prepotential. The holomorphic symmetric tensor can be found from Missing Equation
It can be shown that the curvature related to the KŠhler metric satisfies the relation (3.5) with this choice of.
This formulation is also very convenient to formulate the BPS bound. We can view the charges Q as section in the integral bundle dual to. This is because the charges relate in a linear way a vector of integer numbers to the vector of gauge fields, and the vector of gauge fields now take naturally values in if we include the duals. Using the formula for the central charge (3.18) we can write the bound
Missing Equation
One can find non-projective special coordinates by dividing out one of the homogeneous coordinates, say. This gives special coordinates
Missing Equation
The prepotential which is homogeneous of degree 2 should be divided out by. We will write for this function and will also refer to it as the prepotential. Analogously for the dual special coordinates we write. Remark that the relation
Missing Equation
is valid in special coordinates. The dual coordinate can not be derived as a derivative as we lost the dependence of. But from the homogeneity of F and the definition of we derive Missing Equation
The special coordinates can also be seen as a special gauge for the KŠhler-Hodge symmetry on. which is determined by setting to 1. The components of in this gauge are thus
Missing Equation
In this gauge the KŠhler potential assumes the form
Super Yang-Mills theory is a rigid version of the gauge sector of supergravity. It can also be seen as a limit of this sector. To be precise it is the limit of to infinity as this parameter sets the scale of the gravitational coupling with respect to the gauge couplings. Because of this we should expect that also the parameterization of N=2 super Yang-Mills described in the last chapter can be viewed as some limit of special geometry. This is the case, and in fact we can describe the moduli space of this theory with a description much the same as special geometry, called rigid special geometry. We have already seen some of it, because the combined coordinates of the last chapter are in fact special coordinates of rigid special geometry.
In the rigid case the KŠhler-Hodge structure is lost, so that we lose the homogeneous structure. This makes rigid special geometry even simpler than special geometry. For a gauge group of rank n, the vacuum expectation value of the Higgs field is an element of the Cartan algebra of the gauge group. Therefore the moduli space is n dimensional (which is the dimension of the Cartan algebra). So special coordinates are now described by n coordinates. In addition we have n dual coordinates. which are a generalization of the dual coordinate in the case. Furthermore the dualities now act as transformations, generalizing the of the rank one case. These dualities are generated by linear shifts of the dual coordinates, electric-magnetic duality transformations on the components and Weyl transformations on the coordinates and the dual coordinates separately.
In accordance with the construction of special geometry we now define rigid special geometry in terms of a holomorphic symplectic vector bundle of rank 2n over the moduli space provided with a compatible Hermitian metric. We do not have the special line bundle in this situation, so we will require a holomorphic section directly. The components of can now already be directly identified with generalizations of the coordinates and its duals. The KŠhler potential is now defined as the inner product
Missing Equation
This gives the right expression for the metric on the moduli space when we go to special coordinates, and therefore by invariance must generally be valid.
Special coordinates are now defined in the same way as for special geometry, where the flat symplectic basis now consists of 2n sections. We write in this case
Missing Equation
We have special coordinates on the moduli space if the are good coordinates on the moduli space. The are now the dual coordinates. In special coordinates we can still derive these from a holomorphic prepotential. which now has no homogeneity restriction.
With respect to these coordinates the KŠhler potential is
Missing Equation
the second derivative of which gives exactly the KŠhler metric.
This formulation is very convenient to identify the rigid limit of special geometry. The rigid limit can be characterized by the fact that the couplings to the graviphoton are weak. These couplings are determined by the derivatives of the prepotential of the supergravity theory. Therefore we can identify the rigid limit as the limit where the dual special coordinate is large compared to the other components of. Also it must be large with respect to its derivative. In fact it can be compared to the Planck mass level, as this is the level at which gravitation decouples. To keep this situation, we should only consider duality transformations which do not mix with the other components. This means that the basis element is preserved. An transformation S having this property preserves the coordinate
Missing Equation
and in fact do not mix with the other coordinates. Therefore the special coordinates of the special geometry transform simply by linear duality transformations. So it seems that we can simply identify these coordinates with the special coordinates of the limiting special geometry. This is also in accordance with the formula for the KŠhler potential. In special geometry the KŠhler potential is given in the limit. taking the gauge. by
Missing Equation
where is the KŠhler potential of rigid special geometry. As we take large compared to its derivatives we may consider it constant in this formula. Therefore we see that to lowest order in the limit of large Planck mass is given by. modulo a normalization constant. Remark also that a constant shift of K is immaterial because only derivatives of K contribute.
This reduction can also be used the other way around. This gives a simple derivation of the transformation of the prepotential in rigid special geometry. We can - at least locally - always define a special geometry prepotential F(X) from a rigid special geometry prepotential by introducing the coordinates as homogeneous generalizations from and setting
Missing Equation
This function then is automatically homogeneous of degree 2 in. The special geometry related to F now transforms under the rigid transformations embedded in. which preserve and. This implies that for rigid special geometry is always preserved by the duality transformations. This function can be found from (3.38), which now defines in rigid special geometry. Hence we obtain. from which we derive the transformation law
Missing Equation
where is related to by the linear symplectic transformations. Therefore we find that in general transforms with quadratic shifts. The invariant combination in rigid special geometry that we call plays an important role there, just because of its invariance. This was first appreciated in [28]. It is shown there that it is roughly equal to the gauge invariant modular parameter u. Therefore this really characterizes this parameter in terms of rigid special geometry.
4. Heterotic strings
Ę
The heterotic string compactified on a manifold of the type gives rise to an effective 4 dimensional theory with N=2 supersymmetry. We will consider this theory in this chapter. We will look at the perturbative corrections to vector couplings in these theories. This is quite appropriate for this theory because the heterotic string allows non-abelian gauge fields and there is a natural expansion parameter given by the dilaton, which couples to these vector bosons.
We will mainly be concerned with the moduli space of these vector multiplets. We will start with the classical description of this moduli space. After that we will consider some generalities about perturbative corrections to this space and then give a way to calculate these corrections for a large class of models. This will give rise to certain general properties of the perturbatively corrected moduli space, such as points of symmetry enhancement.
We now consider the heterotic string compactified on the space. The effective field theory in 4 dimensions has N=2 supersymmetry and has therefore a moduli space which locally has the form. We concentrate again on the vector multiplet moduli, which can be described with special geometry, where the precise form depends on the gauge group.
The target space supersymmetry of this model is inherited from the right moving, supersymmetric conformal field theory on the internal manifold. The supersymmetry transformations are generated by the zero modes of the gravitini. The corresponding vertex operators are build from certain primary operators in the internal CFT. The N=2 superconformal algebra which these vertex operators have to satisfy, imply certain commutation relations for these primary operators [29]. For the heterotic string, where both gravitini come from the same sector, it is found that the right moving c=9 superconformal algebra must split into a free c=3 piece with N=2 supersymmetry and an interacting c=6 piece with N=4 supersymmetry. The c=3 theory can be described by a free complex superfield, which can be identified with the right moving part of the CFT. The c=6 theory can be identified with a superconformal field theory on K3. This in fact always gives rise to N=4 supersymmetry, as the K3 CFT contains an Kac-Moody algebra which promotes the N=2, which exists on every Calabi-Yau, to N=4. The left moving CFT is unconstrained, except for the fact that it must have c=22 and it must obey modular invariance. To make a direct interpretation as a compactification it may contain a CFT on. although this is not necessary for general N=2 heterotic strings in 4 dimensions. For the moment we shall however take this to be the case.
The Green-Schwarz anomaly in the heterotic string theory
Missing Equation
gives a restriction on the gauge bundle. By acting with the d operator we get
Missing Equation
where R is the Riemann curvature, which must be viewed as a gauge connection here (in this identification it is often referred to as the spin connection), and F is the Yang-Mills field strength. As H must be globally well defined we see that dH is in fact trivial in cohomology. This means that the cohomology class of and must be equal:
Missing Equation
This is equivalent to the vanishing
Missing Equation
for any 4 dimensional submanifold M of the target space. Let us now concentrate on the internal manifold. especially the K3 part as this will give us some important restriction. The cohomology class on a 4-manifold M is a well known topological invariant, and is in fact related to the Euler class of the tangent space
Missing Equation
We will also denote the Euler number of a 4-manifold, which is defined as the integral of the Euler class over the manifold, by. In general the class of is related to the so-called first Pontryagin class by
Missing Equation
where V denotes the gauge bundle. The integral over a 4-space of this class is always an integer. It is well known in physics and is called the instanton number. as it characterizes non-trivial instanton solutions of the gauge field. So we get the relation
Missing Equation
This should be true for any 4-manifold, so also for the K3 manifold. The K3 has a non-trivial Euler number:. Therefore we see that the gauge bundle must have a non-trivial instanton connection, with instanton number 24. So if we want to compactify the heterotic string on a K3 manifold we are forced to use instanton bundles. To also have the rest of the field equation right we can choose the 3-form H zero as we did in the uncompactified case. To achieve this we get a very nontrivial equation
Missing Equation
There is a useful trick to satisfy this equality. Because of the Calabi-Yau condition on the K3 we know that we can identify the spin connection R with an connection, as the piece is trivial. We can now identify this group with a subgroup of the gauge group or of the heterotic string. After that we can solve the equation just by equating the gauge connection for this subgroup with the spin connection of the tangent bundle. This procedure is often referred to as embedding the spin connection in the gauge group. This procedure breaks the gauge group to a smaller group. The precise gauge group of course depends on the details of the embedding of the spin connection.
The most important thing to know is how the is embedded in the gauge group. For the string there is a maximal embedding. referred to as the standard embedding, which breaks one to. This yields an unbroken gauge group where the factors come from the torus, the graviton and the dilaton. Apart from the vector multiplets, which give the gauge group, there are also hypermultiplets present. These come from the moduli of the K3 and of the broken part of the gauge bundle. For the standard embedding this gives 10 charged hypermultiplets in the 56 of the gauge group and 65 neutral hypermultiplets. From these neutral hypermultiplets 20 originate from the moduli of K3, the other 45 are moduli of the gauge bundle. The scalars of the vector multiplets can have a nonzero vacuum expectation value thereby breaking the gauge group generically to by the Higgs mechanism. Also the charged hypermultiplets will become massive as a result of this Higgs effect. Therefore we keep 18 vector multiplets and 65 hypermultiplets. In the general case we can embed the spin connection in many different ways in the gauge group. The number of hypermultiplets and vector multiplets that arise in the various different situations can be calculated using index theorems, see e.g. [30]. In general we thus have a gauge bundle over. We split this bundle into an instanton bundle, which now has no gauge dynamics anymore, and a surviving gauge bundle which can be described by free string degrees of freedom.
We shall now have a further look at the classical moduli space of the N=2 heterotic string compactifications described in the last section. The vector moduli space is determined by the part of the internal manifold and the remaining free piece of the c=16 left moving algebra. Also the dilaton of the heterotic string is in a vector multiplet. This dilaton has a moduli space. Therefore the classical moduli space is given by
Missing Equation
where is the Narain moduli space of (1.64), which describes the part of the moduli space of the heterotic compactification on the torus introduced in section 1.7. We will write for the homogeneous space which is the cover of the Narain moduli space.
There is a convenient way to describe the Narain moduli space in this situation. Therefore we use the general parameterization of the moduli with the map. In this case the right moving space is 2-dimensional. Therefore we can identify the real right moving momentum space with the complex plane. This makes the moduli a complex linear function on the lattice or differently said an element of the complexification. Remark that this is in accordance with N=2 supersymmetry, as this demands a complex structure on the moduli (and even holomorphicity). We will choose a complex metric on the (linear) momentum space. This means that we can write for the inner product. where the scale g is defined by this relation. Using the relation of the metric on to the moduli given in (1.62) we find a restriction on the moduli Missing Equation
The scale of the metric is determined by the inner product
Missing Equation
This gives a further restriction
Missing Equation
on the moduli, which is necessary to have a positive definite metric on the momentum space. This restriction is in fact just the restriction on the KŠhler cone which we encountered before for Calabi-Yau compactifications. Using this and the fact that we can choose arbitrary within the restrictions, that is we can solve for the geometrical quantities G, B and A from section 1.7 for any which satisfies the restrictions, we can parameterize the Narain moduli space locally by
Missing Equation
where the quotient by divides out the homogeneous scalings of the moduli, which are not relevant as mentioned earlier. In the following we will assume that we can split off a hyperbolic lattice H, which is the lattice of signature (1,1) where we will take the inner product for definiteness. Remark that this is generally possible for the models that we discuss here as this lattice is contained as a subfactor in the lattice of the compactified string. So we write the full lattice as. With respect to this splitting we can conveniently parameterize the Narain moduli with complex parameters by Missing Equation
Remark that the choice of 1 for the first coordinate is just a convenient gauge choice for the quotient action by and that the holomorphic restriction on above then fixes the second coordinate. The positivity condition on the moduli now becomes in terms of y
Missing Equation
We see that the moduli space locally has the structure of a cone. This is of course just the KŠhler cone structure we encountered before. The cone in the imaginary hyperplane given by the light cone condition above consists of two parts: a forward lightcone and a backward lightcone. We shall restrict y to the forward lightcone. This is not really a restriction as the two cones are related by a duality transformation, which we have not yet divided out. To have a definite model for the vector moduli space of some supergravity Lagrangian we need a special geometry structure on the moduli space. Therefore we have to find the symplectic bundle and the line subbundle. We have parameterized the moduli space by the pair where and. We will have to give some identification of these coordinates in terms of components of the section of. Remark that this can not be done directly for all the coordinates as S is not a homogeneous coordinate as should be the case for the components of. But it has a natural interpretation as some scale parameter. On the other hand the coordinate is a homogeneous coordinate. Therefore we should expect the complexified dual lattice somehow in the bundle. It turns out that we should take for the fibre of this bundle the vector space
Missing Equation
Remark that this bundle has a natural complex structure and furthermore a natural symplectic structure, as required for special geometry. This symplectic structure is determined by the canonical pairing between and
Missing Equation
Also the dimension is right, as the number of vector multiplets is (remember that the dilaton of the heterotic string is also in a vector multiplet). The intersection form Q on the lattice gives us a natural identification of this lattice with its dual. We can use this to define the line bundle. We will identify this bundle by giving the holomorphic section which we define by
Missing Equation
This section of the bundle is homogeneous, and satisfies the requirements, which can easily be checked. For example the first derivatives of with respect to the moduli are orthogonal to by
Missing Equation
It follows in the same way that all the other derivatives are orthogonal to. Especially the couplings vanish classically.
It is now easy to determine the rest of the structure on the moduli space from the general formulae. The KŠhler metric on is determined by the KŠhler potential
Missing Equation
The KŠhler potential breaks up into a part on the dilaton moduli space and a part on the Narain moduli space. We will denote the KŠhler potential on the Narain moduli space in the following. Therefore these factors of the moduli space are orthogonal to each other at the classical level. In the following section we we will see that this will not remain true when we include higher loop corrections. In fact this is a consequence of a one-loop mixing of the dilaton and the other moduli.
It is often also useful to have some kind of special coordinates. For this we need to order the components of into a vector. A possibility is to take for the the homogeneous coordinates of. and therefore for the dual coordinates. Often this is a convenient choice, but there is a drawback as the coordinates are not independent because they satisfy the relation and therefore are no good coordinates on the moduli space. Related to this is the vanishing of the prepotential calculated from these coordinates. Therefore it is more natural to choose different coordinates. We can obtain good coordinates by a symplectic transformation
Missing Equation
not effecting the other coordinates. From this and the parameterization (4.14) we derive special coordinates
Missing Equation
We have already seen that the first coordinates serve as good coordinates on the moduli space. With respect to these coordinates the prepotential is given by Missing Equation
The bundle of the special geometry is not trivial even in the classical situation. This is because the T-duality transformations induce non-trivial monodromies in the bundle. These monodromies can already be read off from the explicit form of this bundle. A T-duality transformation acts on the lattice with linear transformation T. On the dual lattice it then acts with. Therefore it acts on the fibre of with the block diagonal matrix
Missing Equation
The singular points in the classical moduli space are the fixed points of the duality transformations. These correspond to the enhanced symmetries. A monodromy around such a singularity corresponds in the covering space with a path connecting endpoints related by T-duality transformations. In the covering space the corresponding symplectic bundle is trivial, so transport along this curve is also trivial. On the other hand in we have to identify the endpoints with the T-duality transformation, which also acts on the fibre of as above. So this must be the way a monodromy acts in the bundle. On the section it acts then as Missing Equation
These transformations are symplectic transformations in the fibre of the bundle. This is required as the monodromies must give equivalent theories, just as we had in the rigid case. Like in the rigid case we can expect more complicated monodromies when we include quantum corrections. Especially we could have symplectic transformations which mix the coordinates and its duals. These can however not be seen at the classical level. In the following we will see these transformations, but these will still be perturbative. The general non-perturbative dualities are much harder to analyze and require an approach like Seiberg-Witten theory, but now for the full quantum gravity.
A special role is played in the heterotic string by the dilaton. As we have seen the VEV of this field plays the role of coupling constant, or more precisely the exponent does. It turns out that in N=2 heterotic string theory this field naturally forms a complex field together with the axion field a which is dual to the B-field. That is, the axion is related to the Kalb-Ramond field by
Missing Equation
where * denotes the Hodge-dual in four dimensions. This complex field is called the axion-dilaton field (we will often refer to it simply as the dilaton). This field is really the scalar field (modulus) of a vector multiplet in heterotic theories. This means that the prepotential depends holomorphically on this complex field. The perturbative prepotential can be written as an expansion in negative powers of the axion-dilaton, because the dilaton serves as the loop counting parameter. At tree-level the dilaton comes in linearly, so that the g-loop contribution must be proportional to. Remark that this is in accord with the general counting argument in the string theory (1.6). In string theory we know that there is a perturbative Peccey-Quinn symmetry which acts on the axion by shifts. This is the dual version of the shift of B by integral 2-forms, which is an exact symmetry because it changes the action by an integral. This transforms S by. The classical Lagrangian is symmetric under the continuous symmetry, but in string theory and also at the non-perturbative level in supergravity this symmetry is broken to a discrete symmetry with. In any case, this symmetry does not allow negative powers of S in the Lagrangian (and therefore also not in the prepotential). Therefore there can only be one-loop corrections to the perturbative Lagrangian of the heterotic string. This also shows that the non-perturbative corrections are proportional to. where. We can thus write the prepotential as
Missing Equation
and the corresponding relation for. From this general form we can derive how the mixing of the dilaton with the Narain moduli occurs, in terms of the perturbative correction.
The couplings of the (non-abelian) semisimple gauge factors in the gauge group are given at tree level by the universal formula
Missing Equation
where the factor is the level of the Kac-Moody algebra related to the gauge group factor.
In general there are perturbative corrections to the universal part of the coupling so that we have in string theory the universal coupling at one loop Missing Equation
where denotes the universal part of the threshold correction. So now we obtain from (2.37) for the one-loop renormalized couplings the following expression Missing Equation
For supergravity there is another way to write the coupling at one-loop. This makes use of the Wilsonian coupling. The Wilsonian coupling is the coupling that emerges in the effective Lagrangian after integrating out the heavy modes, above the scale. The renormalized couplings we have studied until now are the couplings corresponding to the 1PI diagrams. The Wilsonian coupling should exhibit explicitly all the symmetries of the theory which do not involve scales. Especially in N=2 supergravity the complexified Wilsonian coupling is a holomorphic function of the moduli by the supersymmetric Ward identities mentioned before. In [19] the renormalization of the Wilsonian couplings are calculated for N=1 supergravity. With the relations of the field renormalization to the KŠhler potential in N=2 supergravity the renormalized couplings can be written [35] Missing Equation
where denotes the Planck mass, which serves as the cut-off scale for the Wilsonian coupling in string theory. The true unification scale is related to this scale by. This can also be seen from a comparison of the dilaton dependent terms in (4.31) and (4.32).
The threshold correction can be calculated from string theory. To do this, we calculate the -angle for the gauge group factor. We therefore calculate the CP-odd three-point function of two gauge fields and one modulus field which we denote T, in one-loop order. From the effective Lagrangian this should be
Missing Equation
where is the formal derivative of with respect to T. We have only a formal derivative because it turns out that there is a holomorphic anomaly, so that this term is not integrable. There is however an N=2 Ward-identity which states that and this relates the diagram to the threshold correction, because the tree-level coupling is independent of the modulus T, so we have in fact. This relation is also valid at one-loop provided we take for both the complete one-loop result, including the reducible diagrams. We will come back to this point later.
This calculation is performed in detail in [31][32]. From this we find for the threshold correction to one-loop order Missing Equation
with Missing Equation
In this formula the trace is over the Ramond-Ramond states of the internal theory, F is the fermion number operator. Only the RR states contribute because the diagram we calculated is CP-odd, therefore only the odd spin-structure on the worldsheet is to be considered. This also is responsible for the insertion. The term with is in fact universal and could therefore have been included in the universal threshold correction. But with this combination the function is modular invariant. The subtraction of in (4.34) is included to remove infrared divergences of the integral. These are the same as for the effective field theory, so also proportional to. and hence can be included in the -dependent term.
A similar formula also can be derived for gravitational couplings, which are loop corrections for two-graviton interactions. It turns out, [33] and [34], that the resulting formula only differs from (4.35) in that the charge insertion is replaced by. These corrections can be calculated with the same techniques that we will describe shortly for calculating the threshold corrections.
The universal part of the coupling,. can also get one-loop corrections. We already anticipated this in (4.30). In fact when we define the one-loop threshold corrections as the corresponding string amplitude as above, a one-loop correction must be included to account for the mixing of the dilaton and the other moduli at one-loop in string theory. This can be seen from the one-loop corrections to the KŠhler potential which only depend on the moduli and not on the dilaton, as the latter serves as loop counting parameter. We write the KŠhler potential at one-loop as
Missing Equation
Here is the moduli dependent (but not S dependent) tree-level part of and is the Greene-Schwartz term, the one-loop correction to the dilaton. This is by supersymmetry related to the Greene-Schwartz anomaly for the gauge fields. This term therefore determines the one-loop mixing of the dilaton and the other moduli as the KŠhler potential determines the 2-point functions.
We can now relate the universal threshold correction to the function which determines the one-loop correction to the dilaton propagator. For this we use the fact that the one-loop string amplitude calculates the full amplitude at one-loop, while the renormalized coupling represents the 1PI diagrams. Therefore we must have the equality in figure 4.1, which compares the string loop diagram and one-loop corrections in the effective field theory. Missing EquationĘ
Figure 4.1: Comparison of the one loop corrections to the coupling from string theory to those of the effective field theory
In this figure the external line is a scalar modulus other than the dilaton, which we again denote T. The dashed line should be the dilaton. as this is the only field that couples to the gauge fields at one-loop. The string diagram is by our definition determined by the threshold correction. To be precise, it must be a derivative with respect to T of this correction. This follows from Taylor expansion of the gauge-kinetic term in the effective action. After dividing by the momentum dependent factor the relation above can be written
Missing Equation
Here we used the tree-level propagator for the dilaton, which is given by the KŠhler metric on the moduli space. Comparing this expression to (4.31) we see that apart from a possible moduli independent constant the universal coupling constant must be given by
Missing Equation
Using the formula (3.40) for the prepotential in special coordinates we find at one-loop
Missing Equation
From this result we derive the Green-Schwarz term describing the one-loop mixing of the dilaton as
The moduli fields transform under the perturbative T-duality group which for this situation is. As the Narain moduli have a fixed relation to perturbations in the string theory, the duality transformations on this fields will not be modified at the higher loop level. The transformations on the dual coordinates will be modified in general.
From the fact that the moduli are not effected by quantum effects it is convenient to make use of the parameterization induced from. although a prepotential does not exist. In general we will write these coordinates as. where at tree-level is given by (after the identification of the dual lattice by Q). The one-loop correction to is given by. where denotes the derivative with respect to the coordinates of which F is a function. This is true also for these coordinates because we have.
We have seen in (4.26) how monodromies associated to T-duality transformations act on the classical moduli space with block diagonal symplectic transformations in the bundle. When we include the one-loop corrections to this will no longer be true. As for the rigid case the will have shifts which must be linear in the to be in accordance with the symplectic transformations. So in general the perturbative transformations (4.26) are modified to Missing Equation
This transformation must be a duality transformation and therefore it should be symplectic. The corresponding symplectic matrix is
Missing Equation
The symplectic condition restricts C to be a real symmetric matrix.
Also the prepotential is not invariant. The transformation can be derived from the transformation above. To see how it acts observe that we also in this coordinates have the relation
Missing Equation
as the tree-level inner product vanishes and the one-loop correction to vanishes. From this we see immediately Missing Equation
So F transforms with quadratic shifts in the moduli. This transformation property is equivalent to a logarithmic behaviour of given by
Missing Equation
near the fixed point. Calculating the perturbative corrections then determines C, which can be read of from the singularities.
The dilaton will get nontrivial transformation properties as it is related to a dual coordinate in these coordinates. However we can define a modified dilaton field by Missing Equation
Although S has a very nontrivial transformation we get for this modified field the transformation property (cf. [35]) Missing Equation
where the trace is with respect to the intersection form on the moduli lattice. This transformation can be derived as follows. Consider a modified prepotential which is now a function of S and extended Narain moduli. so without the restriction. We define this analogously by in such a way that the modified components still transform as before, only writing instead of. It follows that also the prepotential transforms analogously. Remark now that at tree-level we can get the dilaton S by taking a second derivative
Missing Equation
We take this as a motivation to define the modified dilaton as in (4.46), as this can be written as the limit of taken at (so satisfying the restriction) of
Missing Equation
The transformation of in this formula follows easily and is given by (4.47). So transforms with constant real shifts. Remark that is not a special coordinate, as S is. In some cases it is possible to modify the dilaton even further to obtain an invariant modulus. We will not do this here because this only complicates things unnecessarily.
Using the modified dilaton field we write the Wilsonian coupling occuring in (4.32) at one-loop as
Missing Equation
where is a holomorphic function of the Narain moduli. Because the physical couplings must be invariant under the perturbative dualities and the transformation with real shifts of. we find that the combination
Missing Equation
must be invariant. As is a holomorphic function of the moduli this comes down to the fact that must be a holomorphic modular form of weight for the T-duality group. The weight follows from the fact that is quadratic in the moduli and therefore transforms with weight two. On the other hand the singularities of are only allowed to lie in the singularities of the moduli space, that is the fixed points of the duality group in the cover. These properties in fact determine up to a constant factor.
What we have seen is that the prepotential in general must have certain automorphic properties with respect to the T-duality group. This point has been stressed much in the literature, see e.g. [36] and [37]. There this property is generalized using the conjecture that the automorphic property will remain also with respect to the non-perturbative dualities. These are the dualities that are not upper diagonal. To get this one should include also magnetically charged and dyon states in the calculation. We will not go that far and only consider perturbative corrections using the one-loop calculation (thus involving only perturbative, electrically charged modes) discussed before. It will turn out that the resulting contribution to and as defined above have the nice automorphic properties that are expected (remark that this should be true for by definition).
The Virasoro constraint on a physical state imply that the mass of such a state is given by equation (1.34), which for the heterotic model under consideration can be written
Missing Equation
where is the right moving oscillator level, is the conformal weight (zero mode contribution) of the right moving part of the state and is the right moving momentum of the remaining right moving torus.
BPS states must in addition satisfy the constraint. The central charge of the super-PoincarŽ algebra is determined by the charges of the gauge fields in the right sector, where the supersymmetry originates from in the heterotic string. These charges are the zero modes in the right sector which properly normalized are precisely. so that the BPS condition becomes
Missing Equation
Therefore there are two kinds of BPS states: those with and and those that satisfy and (the fermionic oscillators contribute half integers to the level and the conformal weights). We refer to these different cases as vector and hypermultiplets respectively, although this is not quite what they represent in the effective space-time theory as this classification describes only the right moving part.
We have seen that the couplings in the effective string theory are related to a modification of the supersymmetric index defined by Missing Equation
This supersymmetric index is really some generalization of the elliptic genus of the manifold provided with a gauge bundle.
The Hilbert space of the superconformal theory can be decomposed using the splitting of the superconformal algebra. Using also that any state is in a representation of the algebra of the K3, we have
Missing Equation
where the upper index represents the central charge and the lower index refers to the right moving zero mode quantum numbers. The quantum number l denotes the highest weight of the representation. The left moving Hilbert space depends on these quantum numbers through the level matching condition which the physical states must satisfy.
The current which appears in the supersymmetric index is given for the heterotic string by
Missing Equation
where is the current of the c=3 algebra and is twice the current from the Cartan algebra of the inside the c=9 algebra. We can now split the trace in (4.54) according to the Hilbert space above. It then also contains a trace over the representation l. As and do not vary inside this representation, the only relevant factor of this trace is Missing Equation
as the trace with the insertion vanishes for any representation l. In fact using representation theory of the N=4 superconformal field theory we can say something more about this trace. All massless states correspond to short representations of the N=4 algebra, which consists of the primary states. In the Neveu-Schwarz sector we have for these states the restriction h=q=2l, while l is restricted by the level of the Kac-Moody algebra to. Therefore the states in the NS sector are given by (0,0) and where we label the N=4 states by their conformal weight and -spin (h,l). Remark that these are precisely the states we found earlier by the BPS-condition. In the Ramond sector these are related via spectral flow to and respectively. These are, as should be, exactly the primary states which satisfy. The traces (4.57) for these states are determined by the so-called Witten index on the K3 theory
Missing Equation
These states are the only representations for which the Witten index is nonzero. This is because the massive representations contain several values of l, whose contributions cancel each other. This implies that the supersymmetric index (4.54) only depends on these representations thus it is a sum over BPS states. So we see that the threshold corrections are only determined by these finite number of string states, and do not depend on the whole infinite tower that exists. This is a great simplification.
We will denote a representation of the full superalgebra by their conformal weights and their charges as. where and q denote the conformal weight and charge of the c=3 algebra on the and and l denote the conformal weight and the charge of the c=6 algebra on K3. The primary Ramond states of the N=2 algebra must satisfy. Therefore the relevant BPS states in the Ramond sector are given by and. These two options correspond to states which are vector multiplets and hypermultiplets from the internal point of view respectively. The latter state with l=0 occurs twice in any hypermultiplet, therefore it should be counted twice in the trace. We can now write the trace over a multiplet for these two possibilities as
Missing Equation
Therefore we can interpret the full partition sum (4.54) as Missing Equation
where as mentioned before the terms 'vector' and 'hyper' multiplet should be seen in the context of the internal theory only. This sum is to be taken over the complete set of BPS states, and not only the physical ones. For example the tachyon, which has negative mass and is not a physical particle in the string theory, gives a contribution. This latter contribution is now seen to be (it is counted as a vector multiplet in our conventions). Using the splitting of the gauge bundle into an instanton bundle over the K3 manifold and a remaining free bundle we can split the trace in (4.54) into a trace over the K3 instanton bundle and over the remaining free bundle. Only the latter part depends on the moduli as the K3 moduli do not couple to the vector moduli. Therefore we may write the trace as Missing Equation
where is the partition sum of the zero modes of the free gauge bundle
Missing Equation
which contains the moduli dependence, and is the elliptic genus of the K3 manifold coupled to the instanton bundle. It is just defined as the trace over the K3 and instanton sector of the theory. It is purely topological so does not depend on the moduli of the K3, so it can be calculated in any simple model for the K3 such as the orbifold realization. Because of the modular invariance of the full partition sum it follows that G(q) has modular weight -r/2. Furthermore it is a meromorphic function of the modular parameter in general.
The true threshold correction we are interested in has an extra insertion related to the gauge group generators. This modifies the trace to Missing Equation
Remark that although is still modular invariant it is no longer a holomorphic function of. Therefore the relevant 'elliptic genus' is now of the form Missing Equation
where the are holomorphic modular functions of the proper weight and is the modular covariant derivative of Appendix B of weight w=-2-r/2, which is the proper weight of. Remark that although is not modular the combination is a modular function. The equivalence of the two expressions is explained in Appendix B. Note that the modular form is universal because it only depends on the model and not on the gauge factor. In fact it is equal to the corresponding factor in the supersymmetric index in (4.61). The complete dependence of the gauge factor is in the modular form (or ). We will see that precisely the part with corresponds to the universal part determined by the perturbative prepotential.
We will in the following use expansions of these modular forms
Missing Equation
The differential part of the covariant derivative acting on F then gives an expansion of the form Missing Equation
where the first equality defines G'.
We are thus led to calculate integrals of the following type Missing Equation
We will perform this calculation in the next section.
From the above considerations we see that the threshold corrections can be seen as a sum over BPS states, with the charge insertion. This leads, after integration, to a regularized version of the following sum
Missing Equation
This should be clear as the integral over gives approximately the logarithm of. Such sums were considered in [37], where also hypothetical non-perturbative states were included. There a kind of theta-function regularization was used to make the sum convergent. The divergence is however a consequence of the fact that to get the logarithm we integrate over the complete half-line. We should however only integrate over the fundamental domain. in order not to overcount states. Doing this from the start will automatically regularize the sum above. We will see in the next section how this fairly tricky integration can be performed exactly.
We take for the Lorentzian lattice. Here denotes the compactification lattice of the torus and therefore has rank 2. Much of the following however goes through also for a different number of dimensions. The lattice denotes the root lattice for the extra left moving gauge group and is assumed self-dual. Remark that this implies that the rank r is a multiple of 8. To describe the situation for a different rank one must simply restrict some of the directions of the moduli fields to 0. We take the Lorentzian norm (intersection) of the lattice to be
Missing Equation
where C is the self-dual negative of the Cartan matrix of. We take left and right moving momenta satisfying
Missing Equation
The right moving momentum is given by as in equation (1.61). The partition function of the bosonic modes is given by
Missing Equation
where and the matrix H is given by
Missing Equation
We shall now calculate the integrals in (4.67), and thereby determine the threshold corrections, using a method similar to that first used in [38] and recently generalized in [39]. We will use this method in a slightly different way, also making possible some direct generalizations to lattices which are not of the form. This gives a more direct generalization to cases such as the one studied in [40] for. where we have no direct splitting. We will however not go into details in these cases. First of all we consider integrals of the form
Missing Equation
to include the possibility that has signature (d+r,d) for general d. For a general lattice of this signature we should take to get modular invariance. This will greatly simplify the calculations later on. The density of the integral is therefore altered slightly, to get the proper modular invariance properties. For notational convenience we will leave out the infinite term proportional to c(0) for the moment, we just have to remember that we have to subtract this to get a finite answer. Also now we assume the lattice to be of the form where H denotes the hyperbolic lattice with intersection form. where for our purposes d will be 1 or 2. We use again to parameterize the moduli, but with an arbitrary normalization. We shall therefore divide out by this norm.
Furthermore for d=2 we use an identification. which can be done as is two-dimensional. We parameterize
Missing Equation
where now x, z and are real for d=1 and complex for d=2. Here. We assume x>0 in both cases. In the case of d=2 we get furthermore the relation from (4.10). The extra normalization factor is given by
Missing Equation
here we have written when d=2. The partition sum now becomes Missing Equation
where a Poisson resummation on m was performed. We also wrote. where the integer p results from the Poisson resummation. Remark that we can write this as.
In this form modular transformations on the parameter induce linear transformations on the lattice. This can be seen by writing the transformed N as
Missing Equation
In this equation S acts from the left on. inducing a modular transformation. On the other hand it can also be viewed as acting on the lattice from the right with linear transformations.
This can be used to simplify the integration. To see this we first perform the summation on in (4.77)
Missing Equation
Because the partition sum is modular invariant we should have
Missing Equation
This can also be seen directly, using the transformation of N above and the modular invariance relation for even self-dual lattices in Appendix B. Therefore part of the summation on N can be replaced by a sum over modular transforms of. What remains is a sum over -orbits in. These orbits can be represented by elements (p,0) where. Hence the sum over N breaks up into a sum over p (which represents the orbit) and a sum over -transforms of (p,0). The second sum can now be brought over to a summation over transforms of. which can be implemented by integration of over a larger domain. There are two kinds of orbits, namely. which is called the non-degenerate orbit, and p=0, which is the degenerate orbit. For the non-degenerate orbit the domain of integration becomes the complex strip
Missing Equation
We do not obtain the whole half-plane because the subgroup of upper triangular matrices in acts trivially on (p,0) and should therefore be left out in the summation. On the other hand the degenerate orbit consists of one element, so the domain of integration remains the fundamental domain.
Splitting of the degenerate orbit the integral can now conveniently be written (leaving out the index )
Missing Equation
where denotes the integral of the sum over the nondegenerate orbits only.
For the nondegenerate orbit the integral over becomes simple, as the only dependence of the integrand on is in
Missing Equation
so that the exponent of this expression is simply restricted to 0. This is in fact the level matching condition for the string states, which equates the left and right zero modes of the state. So we get, after a change of variable Missing Equation
where d now denotes the dimension of the right moving part of the full lattice. The last equality where we calculated the integral is performed in Appendix B. If the function is an exponential (or a derivative of an exponential not involving p), which will be the case for our integrals, then the sum on p can easily be performed and then yields a polylogarithm. or some derivative. Some care must be taken to account properly for the case. as then the integral is not well defined for. This is a consequence of the fact that in the derivation we changed the order of summation and integration and we must subtract the infinite term proportional to c(0). This is properly handled by including in the answer for an extra term
Missing Equation
with. where denotes the Euler constant.
We begin with the integral. This integral is precisely of the form above with.. x=0 and. So for the nondegenerate orbit we obtain, as is exactly an exponential, a polylogarithm. The remaining integral is now an integral of the same type for the smaller lattice. which can be calculated using the same relation if we can again split of a hyperbolic lattice H. So let us now assume that the lattice is of the form. As for d=1 the moduli are real we have. It has and. The sum on p and the remaining integral give terms which are linear in the moduli.
The complete answer for the integral can be written written in the form
Missing Equation
where we define a function associated to G
Missing Equation
The product is taken over which are positive in some sense, such that they cover half the lattice. The positivity condition does in fact depend on the range in which the moduli y are taken, as it determines the convergence of the product. To be precise, it must be chosen in such a way that for every. Therefore we find different regions or chambers in which the formula is slightly different. We can choose these chambers to overlap, therefore the formula changes smoothly when we cross the wall of such a chamber. The vector is determined by the second integral from the degenerate orbit. It also depends on the precise range of y.
The form is in fact an automorphic form for the duality group which acts on the moduli and is holomorphic in the complete moduli space parameterized by y. This is clear from the product formula if this product converges and therefore the positivity condition for remains the same. That this property stays the same even when y goes to a different range of convergence for the product is a highly nontrivial property, but can be shown for these cases to be true.
We now proceed to calculate the second integral which involves the modular covariant derivative. The integral is still modular invariant, so the analysis of the first integral goes through. Writing. we find two terms: one involving G' from equation (4.66) and a term proportional to. The first term is of the form above with and can be calculated in the same way as the first integral. The second part has an extra in the denominator of the density, and therefore has for the nondegenerate orbit. This implies that the integral now is of the form as explained in Appendix B. Because we must take the derivative can be written. where we suppressed the contraction of and the gradient. Remark that this derivative does not involve p, so we can put the derivative term in front of the p summation. The exponent of p is -3, therefore the sum over p gives a representation of the special function. The degenerate orbits now also give terms cubic in the moduli y, which can be written as
Missing Equation
The terms linear in y are now absent as is the factor. This is because the c(0) does not contribute in the first part because it should be multiplied by n=0. The integral is therefore of the form Missing Equation
where is the Riemann -function, which originates from the contribution in the nondegenerate orbit as. The first term in the expression involving the derivative G' can be rewritten using the identity Missing Equation
Therefore we can combine this term with the second term, which also is a derivative of the same sum.
Using the two different expression for the couplings (4.31) and (4.32) we obtain a relation for the prepotential containing the quantities we have now computed Missing Equation
where the second order differential operator is defined by the first equality. As is effectively determined by its transformation properties and holomorphicity, we can determine the one-loop prepotential from this equation.
Combining the two integrals we now have an expression for the complete one-loop correction to the coupling
Missing Equation
where we explicitly referred to the modular forms we used in the integrals. When we now insert this result in the relation (4.91), we see that we may compare the term including the one-loop correction to the prepotential with the integral and the correction to the Wilsonian coupling with (remember that is universal, while depends on the gauge factor). This is directly clear for the terms involving the (poly)logarithms from the form we derived for these parts of the integrals. So we get two equations
Missing Equation
Remark that precisely the second order operator occurs in the answer for above when we use (4.90). Also the last two terms in (4.89) can be written as the real part of this operator acting on a holomorphic expression. Therefore we obtain for the one-loop prepotential the result Missing Equation
where the modified cubic is determined by. where the ellipses stand for the symmetric completion. This answer is only determined moduli terms in the kernel of the imaginary part of the operator. It turns out that this kernel consists exactly of terms quadratic in (when expressed in terms of y). But this is just the ambiguity in as such terms are changed by the monodromies. Also these are the terms which do not contribute to any of the couplings, including the Yukawa couplings, as these are given by third order derivatives with respect to the homogeneous coordinates.
The modular function in the Wilsonian coupling is given by
Missing Equation
This function has the right modularity properties as we will discuss in the next section. The non-holomorphic dependence, given by the KŠhler potential. matches if we have. This matching is also necessary to get the right weight for as we will see in the next section. This matching really occurs in the cases which have been studied (see [39]). In fact it follows from our general considerations on the elliptic genus. We saw that we can write the elliptic genus as a sum over multiplets. As we saw that the factor has no zero order term we deduce that is in fact the zero order term of the part containing the gauge generator. So we deduce from (4.60), after inserting the gauge generators,
Missing Equation
where we used the explicit expression for. Remember that the matter is formed by the hypermultiplets and the vectors are in the adjoint of the gauge group.
Functions of the form are much discussed in the mathematical literature recently (see e.g. [41]). They turn out to be connected with modular forms of groups like and subgroups thereof. As these are precisely the modular groups for the heterotic string, because these must be subgroups of the T-duality group, it is not surprising that this happens. These groups are also intimately related to possible enlargement of the corresponding Lie algebras called generalized Kac-Moody algebras. They are often the denominator formula of such infinite dimensional algebras, see e.g. [43] and [44]. In string theory Kac-Moody algebras form a well known structure as they turn up in torus compactifications as current algebra related to enhanced gauge symmetries. So it would be very natural to expect such a structure turning up in the string theory. An attempt to find a realization of such algebras in terms of the string modes of the heterotic string has been made in [39]. The construction was however not direct. A much more intrinsic construction was made recently in [42]
We will now discuss the connection of the modularity given in [41] of functions of the form. Borcherd introduces the notation of rational quadratic divisor or RQD, which is the locus of
Missing Equation
in the space for a lattice vector which satisfies. Remark that this condition is also relevant for the heterotic string as states in string theory are related to such lattice vectors. Especially BPS states with are relevant, as these correspond to states which may become massless. Remember that. The mass of BPS states is in addition determined by the central charge, which is given for an electrically charged state with momentum vector (=charges) by
Missing Equation
So the loci of massless BPS states are RQD's in this language. That these are related to the singularities in can also be seen from the expression for I. Near the vanishing locus of there is a singular contribution to I of the form
Missing Equation
In [41] Borcherd has proven that if is a meromorphic modular form of weight -r/2 with integer coefficients and all poles at cusps, then there exists a unique vector such that can be analytically continued to a meromorphic automorphic form of weight c(0)/2 for. Furthermore, all zeroes and poles lie on rational quadratic divisors with multiplicity at the RQD defined by given by Missing Equation
This is precisely what we need for to be modular with respect to the T-duality group. With this theorem it is possible to determine the singularities in the perturbative correction to the Wilsonian coupling. We find now that they are exactly related to BPS states which become massless. These states should be identified with the enhanced symmetries that turn up in the moduli space of the heterotic string. We have seen that these occur at the fixed points of the T-duality group. This relates the singularities in to the enhanced symmetries. We can say even more. Remember that the coefficients c(n) are the degeneracies of the BPS states. So the multiplicity m in (4.102) is equal to the number of (charged) BPS states that become massless at the locus related to the RQD in the moduli space. The logarithmic singularity in and therefore in the coupling, is given from this multiplicity by
Missing Equation
This is, after taking into account the correct normalization, exactly the singularity we would expect when m vector multiplets become massless. Therefore the physical interpretation of these RQD's as enhanced gauge symmetry points seems to be correct.
The prepotential also has singularities associated to these enhanced symmetries. These arise precisely when the Narain lattice contains a (simple) root lattice. A root lattice is here seen as a lattice generated by roots which have norm. These correspond directly to RQD's as remarked above. Let us see what the perturbative singularities at these points are. We expand around a large value of y. We will compare to the rigid limit. Therefore we assume this value of y to be of the order of the Planck mass. Hence we write
Missing Equation
where z is small compared to. which is of the order of. To have enhanced symmetries we assume that there is a root lattice in the kernel of. that is for every root. Then in the expression of the prepotential (4.95), all contributions from are suppressed as is assumed to be large. Therefore we keep in this limit only a sum over. From this sum only the positive roots remain in the sum as for n<-1. Using the asymptotic expression and the fact that c(-1)=2, we therefore obtain the asymptotic expression
Missing Equation
In this expression is a modification of the dilaton, including all contributions from. This is exactly the perturbative result obtained in the rigid limit, which is the generalization of the result (2.42) for a general group.
The prepotential should have monodromies related to the T-duality transformations in the cover of the Narain moduli spaces explained in section 4.3. Using now the explicit expression for the perturbative prepotential in (4.95), we can really derive these monodromies. Remember that these must turn up as shifts quadratic in the moduli and therefore quadric in the moduli parameters y. So let us see what will happen when we perform a T-duality transformation. First of all we may end up in a different region for y, that is a region where the positivity condition is different. This makes that we really get a different expansion. This turns up in the fact that the exponential in some of the terms involving are changed. This change is of the form and can be calculated with Missing Equation
The other change that may occur is that in a different region the cubic terms determined by are different. All these changes are calculable in explicit models (see [39] for an example). From this the monodromies in are directly derived.
Remark that the whole analysis until now was perturbative. Nonperturbative corrections may alter it, and there may even not be nonabelian symmetries present in general. We can even expect this in view of our experience with the rigid N=2 nonabelian gauge theory in chapter 2. There we saw that nonabelian gauge symmetries were never restored, but instead hypermultiplets became massless. We therefore expect the same thing to happen here. The small coupling point at the other end of the moduli space must on the other hand be correctly described by the perturbative analysis. Therefore everything about the small coupling singularities in the heterotic string theory remains correct, including the perturbative monodromies about these singular loci in moduli space.
The simplest heterotic string model we can calculate now is the three moduli model, which has r=0. Thus the only gauge symmetries are the ones that come from the torus. This gives two Narain moduli which are the complex structure and the KŠhler structure of the torus. It is common to use the notation
Missing Equation
for these moduli. They are the parameters in the natural basis for the moduli of the hyperbolic lattice H. The moduli are then given by. The T-duality group for this model is therefore the orthogonal group for the lattice which is. This group is generated by standard fractional linear -transformations on both the T and the U modulus and a T-U exchange symmetry. This reflects itself in the group isomorphism
Missing Equation
This model is already quite interesting to show the properties mentioned above. This model corresponds to the model described before with unbroken gauge group given by. Therefore we will calculate the threshold corrections to the couplings of the unbroken nonabelian gauge group factors and.
The singularities can be found at the enhanced symmetry points, which are given by the fixed points of the modular group. Modulo dualities these are given by the fixed points of the T-U exchange symmetry, and can therefore be found at the locus T=U. At these points one of the symmetries is enhanced to in the classical limit. There are some special points on this locus where the symmetries are further enhanced. These points are fixed by a larger number of dualities. The first point where this happens is T=U=i, where there is an enhancement to. The other point is. where the gauge group contains.
First of all we have to determine the relevant elliptic genera related to these threshold corrections. The partition sum in (4.63) has first of all a contribution from oscillators of the the 24 transverse left moving bosonic degrees of freedom given by. This includes the tachyon and therefore the pole in G. The fermionic contributions are completely regular because there is no fermionic tachyon. The modular weight of G(q) is 0, therefore the fermionic oscillators contribute a modular form of weight 12 to G. This fixes already in equation (4.64) completely. The other modular function is fixed modulo a constant shift, which determines c(0). Remark that this may depend on the gauge factor. The precise result is (see [39])
Missing Equation
Remark that we can write the second modular form also as Missing Equation
where the constant terms in are given by
Missing Equation
These are precisely the coefficients for the two gauge factors. This directly determines the modular form. Remark that the modular form is multiplicative in G. Therefore from (4.112) it follows that factories. The constant term gives functions, while the factor involving can be calculated using (B.9). This determines Missing Equation
This result is completely in agreement with the calculations in [35], which are based on the modular properties of the couplings only.
The prepotential is determined by. In the Weyl chamber it is given by Missing Equation
where (k,l)>0 means or. In the region the prepotential is given by the same expression with T and U interchanged, which is required by T-U exchange symmetry. This function has a branch cut at T=U, precisely at the enhanced symmetry points. The only singularity in at this locus is given by the contribution to the sum of the -term with k=-l=1 and is therefore, using. Missing Equation
This determines already the monodromies around the singular locus. This monodromy can also be calculated using the method explained in the last section. This is a good opportunity to see it in practice. The only difference of in the two regions comes from the contribution k=-l=1. Therefore we calculate using (4.106)
Missing Equation
Remark that these terms can be organized to form a quadratic polynomial in. This allows us to find the matrix C in (4.44) for this monodromy.
A remarkable property of this model is that this already allows us to find the Yukawa coupling explicitly. This nearly fixes the prepotential already. The Yukawa couplings, which are third derivatives of the prepotential, are modular. The transformation of the prepotential is generally of the form
Missing Equation
(in the T-U exchange above we had ), thus under the dualities acting on the parameters T and U it transforms with weight -2 and quadratic shifts. We will calculate the Yukawa coupling, which is equal to a third derivative,. Because of the special relation
Missing Equation
it follows that it is really equal to the third modular covariant derivative. Therefore this Yukawa coupling transforms with weight 4 under the dualities acting on T. Moreover the shift terms are at most quadratic in T, so that the third derivatives vanish. Hence it is exactly a modular form. From (4.116) we see that it has residue at T=U generically. The modularity and this singularity already fixes it to
Missing Equation
The Yukawa coupling for is given by the same expression with U and T interchanged. These modularity and singularity properties were already used in [35] and [45] to find an expression for the prepotential. The latter gives also a more detailed description of the monodromy transformations.
This model may seem special, but really it describes part of all the heterotic models on that we discussed. We took the gauge bundle arising from the -part of the compactification as a free bundle. That is we embedded the instantons in the extra -part of the left moving bosons. In the example above we took the case where the instanton bundle was embedded in an subgroup of one of the groups. If we choose a different embedding giving rise to a two moduli model, the modular function will not change as it is completely set by the modular weight and the tachyon singularity. For the only freedom is in the coefficient of the constant term. which does depend on the model. Therefore only the term depending on this factor may change accordingly. So the formula for the perturbative prepotential is given by (4.115) and can still be given by (4.114).
These formulae are even more universal then just for the 3 moduli models. In all the models used here the moduli include the moduli T and U of the torus. When we set all other possible moduli, which are interpreted as Wilson line moduli for the extra free gauge bundle, we only keep these two moduli. The form of the prepotential in this locus of the moduli space is determined by which is restricted very much by modular invariance. In fact this fixes it completely when. The singularity at infinity given by the tachyon contribution,. fixes the normalization. Furthermore the non-holomorphicity of the integrand in (4.67), determined by the factor. fixes this integral completely. Therefore when we set the Wilson line moduli to zero we find that as a function of (T,U) is universal - at least when r satisfies the restriction above. Hence for these models we find a universal (T,U) dependence of the prepotential given by the three moduli model answer (4.115). This can be written
Missing Equation
where A denote the additional Wilson line moduli and satisfies.
5. Type II strings on Calabi-Yau
In this chapter we consider the type II string theories compactified on general Calabi-Yau manifolds. These string theories also give rise to N=2 supersymmetric effective theories in 4 dimensions. It turns out that in these models we can calculate the vector couplings as a function of moduli exactly, including all quantum corrections. We will review the calculations of these quantum corrections in this chapter.
We will also look at what happens when the Calabi-Yau manifold develops certain mild singularities called nodes. We will see that the direct calculations will give singularities in the effective theory. It turns out that these singularities can be cured by introducing a new light mode in the effective theory. We will show this as far as the vector couplings are concerned and we will see how these extra light modes may arise in the string theory as non-perturbative states.
A Calabi-Yau manifold and therefore also the (2,2)-superconformal field theory defined on this space, is determined by giving the Ricci-flat KŠhler metric and the B-field. Here we have to divide out the diffeomorphism group of the manifold as different pairs (G,B) related by diffeomorphisms give rise to isomorphic Calabi-Yau spaces and -models.
To study the structure of this moduli space we start to look at the tangent space to. We begin with studying the changes of the metric. To eliminate changes in the metric arising from coordinate changes, we require the gauge choice. The requirement that the metric remains Ricci-flat then puts on the condition where here denotes the so called Lichnerowicz Laplacian acting on 2-tensors, which is the natural Laplacian constructed from the Levi-Civita connection. The B field can be varied by adding a closed 2-form to keep H=0, and we may put on a gauge choice. So variations of B correspond to harmonic 2-forms. As harmonic 2-forms are annihilated by the natural Laplacian on forms, which is the same as the Lichnerowicz Laplacian on forms, we can add B and G and identify the tangent space to the moduli space as the kernel of the Lichnerowicz Laplacian.
There is a natural metric on the moduli space which is induced by the -model action, see [46] for details. There are in general two separate kinds of changes for the metric: variations of the complex structure, which are with respect to complex coordinates variations of the form and. and variations of the metric which do not change the complex structure but only the KŠhler metric. We shall refer to the latter case simply as changes of the KŠhler class. It can be shown that these different classes of variations are orthogonal with respect to the metric on the moduli space, and both are orthogonal to changes in B. In physical terms this means that marginal operators related to different classes of variations have no interaction. Remark that changes of the KŠhler class can also be given by variations of the KŠhler form defined as the 2-form. Therefore these combine together with the variations of B into variations of the complexified KŠhler class. The tangent space related to these changes is therefore.
The fact that the KŠhler form is related to the metric puts some non-trivial positivity constraints onto this form, which are given by
Missing Equation
for any 2-cycle C and 4-cycle S. This determines a cone in. which means that if satisfies these conditions then also any positive multiple of does. This cone is called the KŠhler cone and we denote it by. Using the restrictions on we have that the complexified KŠhler class J must be in the complexified KŠhler cone defined by. When we use a proper normalization for the -model action, this model is invariant under integral shifts of B by elements of. The reason for this is that the B-term gives only topological contributions to the action. Therefore we should divide out this group to get the KŠhler moduli space. We can formulate this by saying that B is really an element of. The complexified KŠhler cone should then be viewed as a subset of in this language, as we will assume from now on. Apart from this there are more non-trivial symmetries, such as automorphisms of the Calabi-Yau manifold and duality symmetries which have to be divided out. The variation of the complexified KŠhler form gives what is usually called the KŠhler moduli space.
With this information we have for the moduli space
Missing Equation
where the equivalence relation should contain all equivalences induced by automorphisms of the manifold and any duality symmetry ('T-duality'). It has to be added that although the KŠhler moduli and the complex structure moduli are locally orthogonal, it may not be possible in general to define a corresponding product structure on the global moduli space. It may not even have a bundle structure. This is a consequence of the duality symmetries that are divided out. We will however ignore this subtlety, because we will only work locally in this moduli space.
We proceed to describe the moduli space of complex structures of the Calabi-Yau manifold. In the mathematical literature this is a well known moduli problem and is referred to just as the moduli problem of the Calabi-Yau. The complex structure is determined by a choice of complex coordinates on the Calabi-Yau manifold, moduli holomorphic coordinate changes. This can also be given by the holomorphic 1-forms. or more invariant giving the holomorphic cotangent bundle as a subbundle of. The variation of these forms, and therefore the complex structure, can be given by vector valued 1-forms
Missing Equation
Remark that these forms should be closed, because the differential forms trivially remain closed. The variations are really holomorphic coordinate changes and therefore do not change the complex structure. Exact forms can also be ignored as they represent coordinate changes, which are divided out in the definition of the moduli space. Therefore the only relevant transformations are given by Missing Equation
This gives a canonical identification of the tangent space to the moduli space of complex structures with the cohomology. A more precise description would refer to the variation of the complex structure itself (see Appendix A). Remark however that the holomorphic differentials serve as a local description of. as it is given in local complex coordinates by and its complex conjugate.
The canonical bundle on a Calabi-Yau manifold, which is the bundle of holomorphic 3-forms in the case at hand, is always trivial. This implies that one can always choose a nowhere vanishing holomorphic 3-form. whose cohomology class is unique. The local form of the 3-form is not unique however as it can be multiplied with any non-vanishing holomorphic function. This gauge symmetry is also referred to as the KŠhler-Hodge structure. Although there is no unique choice for it is often very convenient to make some choice. A Calabi-Yau space together with a choice of is called a gauged Calabi-Yau, as it fixes the KŠhler-Hodge gauge symmetry. In fact the choice of inside determines the complex structure because is one dimensional. The moduli space of gauged Calabi-Yau - for the moment forgetting about the KŠhler metric - forms a -bundle over the moduli space of the Calabi-Yau.
Using a choice of. variations of the complex structures can now be identified with differential forms in. This can be done by contraction of the vector indices of the variations, which we already have identified as elements of. as follows Missing Equation
The holomorphic three-form can be used to define the complex structure and will give some useful coordinates in the moduli space. It is always possible to choose a symplectic basis of. This is a basis with the following intersections Missing Equation
With respect to this basis we can give the periods of by Missing Equation
This gives in a natural way rise to special geometry. We can use the first periods as projective coordinates on the moduli space of complex structures. The other periods,. then become functions of the of homogeneous degree 1. These then become naturally holomorphic functions. To make direct contact with special geometry we have to determine the bundle. A point z in the moduli space corresponds to a Calabi-Yau manifold. Both and the variations are in the complex third cohomology of this manifold. We want to identify the holomorphic three-form with the holomorphic section of special geometry. Remember that and its variations span the fibre at a point in the special geometry. Therefore we should choose for the holomorphic bundle the bundle over with fibre at the cohomology class. The intersection form of differential forms gives a natural Hermitian structure on this bundle
Missing Equation
It turns out that with this definition the metric is positive definite. The holomorphic line bundle has as fibre over the vector space. The choice of holomorphic three-form now becomes a section of. as was the case in the general discussion of special geometry. The KŠhler potential is now defined by
Missing Equation
The bundle defined in this way has a natural flat covariant derivative. which in this setting is called the Gauss-Manin connection. The derivative of our section must be in. As both these space are orthogonal to. we see immediately that
Missing Equation
This is called Griffith transversality. With this we have verified the special geometry of this moduli space. Remark that also the splitting of the bundle is immediate here. It is given by the splitting of Dolbault cohomology in the fibre:
Missing Equation
The symmetric tensor can also be given in terms of integrals of forms. It is given in terms of derivatives. Representing derivatives of forms with respect of the moduli with vector valued one-forms. we can write the derivative of with respect to a modulus
Missing Equation
where the ellipses stand for purely holomorphic terms. Remark that this is just the variation defined in (5.5). Going on we find for the third derivative
Missing Equation
where we only wrote out the (0,3) part. From this we find a convenient form for the Yukawa couplings
As mentioned before the type II string compactified on a 6-dimensional Calabi-Yau manifold results in a 4-dimensional effective field theory with N=2 supersymmetry. The (2,2) superconformal field theory on the Calabi-Yau provides the internal sector of the theory and therefore determines the spectrum of the string. We have to distinguish between the type IIA and type IIB string, as they have different 10-dimensional theories.
An important point which makes the calculation in type II strings much easier is the fact that the axion-dilaton S sits in a hypermultiplet. In N=2 field theory there are strong renormalization theorems, one of which prohibits non-gauge couplings between hypermultiplets and vector multiplets. Because the dilaton expresses the loop expansion, we find from this that in type II string theory there can be no loop corrections to the vector multiplet couplings. So the whole calculation for the vector moduli space resumes in the semiclassical or genus 0 sector. As we will see there can however still be instanton corrections from the worldsheet, but also these are restricted to genus 0 and therefore calculable. We will not go any deeper into this.
We will start with a short description of the 10-dimensional theories. They both contain the universal sector of the graviton, the B-field and the dilaton. This is described by the universal effective action (1.76). Furthermore there are in the bosonic sector Ramond-Ramond fields described by p-form field strengths. The difference of the type IIA theory and the type IIB theory is mainly that for type IIA p must be even and for type IIB it is odd. In general the (10-p)-form field strength is just the magnetic field strength related by a duality transformation to the p-form field strength. Therefore we can describe the dynamical degrees of freedom by the field strengths with. Furthermore, the 5-form field strength which is part of the type IIB spectrum is restricted to be self-dual.
For the type IIA theory we get therefore a 2-form and a 4-form field strength, related to a 1-form and a 3-form antisymmetric tensor field. The effective 10-dimensional action of the bosonic fields of the type IIA string is given by Missing Equation
where.
For the type IIB theory we get a 1-form, a 3-form and a self-dual 5-form field strength, corresponding to a scalar and a 2-form and a 4-form antisymmetric tensor field. As the 5-form is self-dual there does not exist an action for the bosonic type IIB theory. What will be important later on is just that the 5-form field strength does not couple directly to the dilaton.
The internal sector is mainly determined by the structure of the chiral ring mentioned in section 1.4 of the internal theory. For a Calabi-Yau model these can be given a geometrical interpretation. We will not be very elaborate here, but will only mention the connection. We will see this connection more detailed in later sections in the slightly different context of topological models. In general these rings can be identified to certain cohomologies related to the geometry of the Calabi-Yau, which are really (commutative) rings. These identification is in general only on the level of linear spaces, not as rings.
The (a,c)-ring is related to the Dolbault cohomology
Missing Equation
The form degree (p,q) is equal to the charges with respect to the pair of -currents in the superconformal algebras. So the grading of the (a,c)-ring corresponds to the grading of the Dolbault cohomology. From the mass shell condition (1.43)and the relation of the conformal weight to the charge (1.41) we find that the (1,1)-forms in this cohomology correspond already to massless states. These are therefore massless scalars in the effective field theory. Remark that these correspond directly to the KŠhler moduli of the Calabi-Yau manifold. In which multiplets these scalars reside depends on the particular string model under consideration. It turns out that in type IIA they correspond to vector multiplets, while in type IIB they are related to the vector multiplets in the effective theory.
The (c,c)-ring can be related to a cohomology similar to the Dolbault cohomology, where the holomorphic cotangent bundle is replaced by the holomorphic tangent bundle
Missing Equation
Now the form degree q is related to the charge with respect to and the vector degree p is related to the charge with respect to J (remark that chiral primary states should have positive charge and antichiral states should have negative charge). So also for the (c,c)-ring the grading corresponds to the natural grading of the cohomology. Now we find that the elements with charge (1,1) correspond to the massless scalars. These correspond exactly to the complex structure moduli as we saw in the last section. Remark that using the holomorphic three-form we can relate the two cohomologies above, as we can change p-vectors into (3-p)-forms. This relation of the rings is a manifestation of the spectral flow which relates the two chiral rings.
The manifestation of the spectrum of the type II string theories on the Calabi-Yau can be seen quite directly from a dimensional reduction of the effective field theory. We will only consider the vector fields, as we will not describe the hypermultiplets. This will give us also the relation of the vector multiplets to the geometry of the Calabi-Yau. In general the universal sectors give the graviton, the B-field and the dilaton on the 4-dimensional space-time. Because there are no non-trivial one-cycles on the Calabi-Yau, we can not reduce them. So they can not give rise to any additional vector multiplet in the effective theory. Therefore all the vectors and matter in the 4-dimensional theory should come from the RR states of the type II theory. We can dimensionally reduce the RR p-form field strengths on cycles of the Calabi-Yau to give rise to several (bosonic) fields in 4-dimensions. This goes as follows. Let C be a (homologically) non-trivial r-cycle on the Calabi-Yau. Then we can define a (p-r)-form in four dimensions by integrating the p-form field strength over this cycle
Missing Equation
When C is a (p-2)-cycle then the reduced form is a 2-form and can therefore be identified with the field strength of a vector field. When r=p then we obtain a scalar. Now the precise relation depends on the model.
We begin with the description of the type IIB theory. There are inequivalent 3-cycles on the Calabi-Yau. Compactifying the self-dual 5-form on these 3-cycles gives vector fields in the effective 4-dimensional theory. With respect to the symplectic basis (5.6) of the third homology we find these vector fields from
Missing Equation
These are really vector fields, but they are related by the self-duality of. These vector fields include the graviphoton. This relation can also be seen as follows. We use a dual basis of 3-forms for the third cohomology of the Calabi-Yau. Then the 5-form field strength can be written as
Missing Equation
This is the only possible way to get vectors as the three-forms are the only odd-degree forms on the Calabi-Yau. Similarly we can get real scalars from dimensional reduction of the 3-form field strength on the 3-cycles. These can be ordered, using the natural complex structure on the cohomology, into complex scalars. One of these can be gauged away using the KŠhler-Hodge symmetry on the Calabi-Yau. This reduction is given geometrically by the periods of in (5.7). From this we see the relation of the vector moduli to the complex structure moduli, as these are related to the 3-form cohomology.
We now proceed with the type IIA theory. This goes much the same way. There are 2-cycles on the Calabi-Yau. Compactifying the 4-form field strength on these 2-cycles gives rise to vector fields in 4 dimensions. This gives the relation alluded to above of the vector multiplets to the (1,1)-forms. There is furthermore one ( ) 0-cycle (a point) on the Calabi-Yau. Therefore the 2-form field strength gives rise to one extra vector field. This is the graviphoton. The scalars are found by dimensional reduction of the 2-form on 2-cycles and the 4-form on 4-cycles. This gives real scalars and hence complex scalars. The scalar of the graviphoton is related to the dimensional reduction of a non-dynamical closed 0-form in 10 dimensions, so that it is naturally non-dynamical from the start.
In general it is hard to calculate couplings in string theory directly. There are however certain sectors of the string for which the couplings can be calculated. These are the sectors which correspond to a topologically twisted model, which is a greatly simplified model which has only a finite number of degrees of freedom. It turns out that these couplings are already enough to determine the relevant part of the prepotential in type II string.
A twisted model differs by the way some fields of the theory transform under rotations. That is, their worldsheet spin is different. The term topological refers to the fact that there is a so called BRST-charge present. This is a global anticommuting operator Q which satisfies. such that the Hamiltonian of the theory can be written as a commutator H=[Q,V] for some operator V. It is then natural to restrict to observables which are Q-closed, that is they satisfy (here denotes a supercommutator, that is a commutator or an anticommutator depending on what kind of operators are inserted). Remark that the Hamiltonian is an observable in this sense. The Q-exact operators, which are the commutators. then decouple from the theory. The space of relevant operators is thus the quotient of these spaces of operators, which is called Q-cohomology. This cohomology is often of finite dimension, which will greatly simplify the calculations. To be able to get this cohomology the BRST-charge Q must be an (anticommuting) scalar. As there are no anticommuting scalars in the original N=2 CFT we are naturally led to a twisting, as then anticommuting fields can become (target space) scalars.
For the type IIA theory compactified on a Calabi-Yau threefold, the relevant part of the conformal field theory on this Calabi-Yau is parameterized locally by the complexified KŠhler cone in. These are as mentioned before related to primary operators in the (a,c)-ring. These are the primary operators in the kernel of and. As the relevant operators in the twisted theory are the Q-closed operators, i.e. in the kernel of Q, to relate these states with the (a,c)-ring we should take Q a combination of these two operators. Hence the twisting should be done in such a way that these operators become scalars, as Q is a scalar operator. Therefore we choose a twist such that the fermions become sections of and are sections of. These fields then naturally combine to a worldsheet scalar section of. The other fermionic fields and become sections of and respectively. The BRST-charge will be the combination
Missing Equation
which is now an anticommuting scalar in the twisted model. Note that we only tensor the fields with powers of the canonical bundle K. As long as we choose for the worldsheet a sphere or a torus with at least two punctures (for the in and the out state) the canonical bundle is trivial. Therefore in these cases the twists of these kinds are immaterial as they only redefine certain fields. Therefore the genus 0 and genus 1 amplitudes calculated in the twisted theory give the same answers as the corresponding amplitudes in the original CFT.
We have now constructed the twisting such that the natural Hilbert space of the twisted, topological theory is the (twisted version of). The relevant local operators are therefore related to differential forms. This relation is given by
Missing Equation
where is a differential form on M. It can be shown that
Missing Equation
so that exact differential forms decouple from the twisted theory. Therefore in this theory the operators can naturally be related to the Dolbault cohomology of M,.
The action of the supersymmetric string can be rewritten in the form
Missing Equation
where J denotes the complexified KŠhler form on M. Remark that as J is closed, this term only contributes to the semiclassical action and has no dynamics. It is quite easy to write out this part. The embedding X defines a linear map. Because the embedded curve is an element of the integer cohomology of M this map preserves the integer structure. The integer cohomology has a canonical basis given by the dual of the class of a point. Let us choose a basis of the integer cohomology. With respect to this basis can now be defined by integer numbers. If we expand the complexified KŠhler form with respect to the integer basis then we have
Missing Equation
The bosonic part of the action is given by
Missing Equation
Because the dynamical part of the action is Q-exact we can multiply it with a free parameter t. The correlations of closed observables can not depend on this continuous parameter, as the theory is topological. This can be seen from the fact that differentiating with respect to t brings down a Q-exact operator, which now decouples and hence gives 0. Therefore we can send t to infinity. In this limit the field X is restricted to be a holomorphic function. Any correlation function becomes therefore a sum over holomorphic embeddings of the worldsheet in the target space M. These are in fact the worldsheet instantons. The possible holomorphic embeddings of the worldsheet are
1. The constant map to a point in M.
2. The map to a simple nontrivial rational curve C, an. embedded in the Calabi-Yau. We shall identify the curve C with its degree, which is the corresponding element of the second homology and which we identify with as above.
3. A multiple cover of a rational curve C. Remark that the degree of an n-fold cover of C is n times the degree of C.
Therefore we can write the amplitudes involving several operators in the twisted theory as
Missing Equation
where is the path integral restricted to the trivial instanton sector and denotes the path integral restricted to the instanton sector corresponding to the n-fold cover of the curve C. We have written and used multi index notation to write the moduli dependent factor.
It turns out that only the product of at most three operators derived from (1,1)-forms give a non-vanishing result. As this is the most interesting case for us, because they determine the ``three point functions''. we shall restrict to this case. The product of operators is, modulo Q-exact terms, equal to the operator related to the product of forms
Missing Equation
Therefore the first factor is completely determined by the restriction to. which must therefore by linearity be given by a multiple of integration over M. We normalize the action in such a way that this multiple is 1. The correlation functions restricted to the non-trivial instanton sectors turn out not to depend on n. Therefore it only remains to calculate the single coverings. They are given by (see [47])
Missing Equation
where X defines an embedding of the curve C. The correlation function between three operators related to (1,1)-forms on M therefore becomes
Missing Equation
where n is the integral degree of the curve C and denotes the number of curves of this degree.
We can now relate these couplings to the prepotential, as given in equation (3.34). We can write the three point functions of the basis elements
Missing Equation
Using (see Appendix B for the special function ) this can now be written as a third derivative with respect to the. As the couplings are also a third derivative of the prepotential we have, up to some quadratic terms in t
We will now look what happens at singularities in the moduli space. These singularities correspond to Calabi-Yau manifolds with a degenerate complex structure. When this occurs the Calabi-Yau manifold is singular as the defining polynomial of the Calabi-Yau and its first derivatives all vanish at the same time. For such choices, some of the periods of the holomorphic three-form vanish. Such a choice results in a singular conformal field theory, but we shall see in the following that string theory can be smoothly continued. Remark that these singular subspaces are generically of complex codimension 1 in the moduli space of complex structures. We will only consider here the simplest case, where the second derivative of the defining equation is non-degenerate. Such mild singularities are referred to as nodes. Calabi-Yau's with one or more of such a singularity are called conifolds and the corresponding points in the moduli space are referred to as conifold singularities. We denote here and below the conifold. In this section we will follow much of the classic discussion in [48] on conifolds.
Using Morse's lemma we can choose local coordinates to describe a neighbourhood of the singularity of a conifold as the vanishing locus of the equation Missing Equation
in. The node is located at. The node locally has the structure of a cone because the equation is invariant under rescalings. To see the topology of this cone it is helpful to separate the variable into its real and imaginary parts. Equation (5.39) then becomes
Missing Equation
The base of the cone is the intersection of the cone with the sphere. which is determined by the equations Missing Equation
The 's in the base form an by the first equation, while the is restricted to for every point. So the base of the base of the cone has the structure of an fibered over an. There exists only the trivial fibration of this form, hence the base must be the product.
In the moduli space, a conifold is some limit of a family of smooth manifolds. We can try to reverse this limiting procedure, and perturb the conifold to get a smooth manifold. There are in general however different perturbations possible which result in smooth Calabi-Yau manifolds of different topology. Locally this can be related to the fact that the node at as described above can either be replaced by an or by an. both of which give a smooth structure. We will now in some more detail describe how these different smoothing operations take place.
We can deform the conifold by adding a term to the defining equation, removing in this way the singular structure. Remark that this simply corresponds to a complex structure deformation, and therefore the conifold is at the boundary of the complex structure moduli space of the deformed Calabi-Yau. For the local description of the deformed manifold we may set this deformation equal to a constant, thus obtaining the defining equation. which is (5.39) in the limit. This means that the node at r=0 in (5.41) is inflated to
Missing Equation
Thus we see that the node is replaced by an. This is schematically indicating in the picture on the right in figure 5.1. We will denote the deformed manifold.
For the other possibility we rewrite the defining equation as Missing Equation
where the entries of the matrix are complex. We still write for these coordinates. The node is at the zero matrix. We now define a smooth manifold by the equations Missing Equation
where are homogeneous coordinates on. These combined equations in define a smooth manifold, as there are no double points. The equation (5.44) implies equation (5.43) because the matrix must be degenerate for a solution to exist. As long as this equation completely fixes z as an element of. therefore identifying this part of the Calabi-Yau with the corresponding part of the conifold. At however z is completely unconstrained and therefore a complete is projected down to the node of the conifold. Hence we see that in the small resolution the node is replaced by a. This desingularization is called a small resolution of the conifold and is denoted. It is shown in the left picture in figure 5.1. The small resolution does not change the complex structure of the Calabi-Yau, as a whole reduced neighbourhood around the node is not affected. It does however add a KŠhler parameter, namely the size of the which is inserted at the node. This KŠhler parameter is zero at the conifold, as then the is reduced to a point effectively. So we can now view the conifold also as a manifold at the boundary of the KŠhler moduli space of the small resolution.
The small resolution of a node is not unique. If we replace the matrix in the defining equation of the resolution by its transpose then we obtain a different manifold with a different topology. The (co) homology is invariant under this transformation, but the intersection matrix of the homology changes. It turns out that the deformation however is unique. Missing EquationĘ
Figure 5.1: Deformation and small resolution of a node of a conifold
When going in the moduli space of a certain class of Calabi-Yau manifolds from a deformation to a small resolution the topology of the Calabi-Yau changes drastically. This follows already from the fact that for each node an is replaced by an. From this follows directly the change in Euler number of the manifold. Let us assume that the intermediate conifold has P nodes. Since and the change in Euler number is given by
Missing Equation
It is possible to say more about the relation between the homologies of the two desingularizations. In the deformation the from the boundary of the small neighbourhood around the inflated makes up together with the generator of the cone a three-ball. indicated in figure 5.1 as the front side of the right figure. Thus we see that in fact the cone with base is replaced in the deformation by. The is really the part of a three-cycle on the Calabi-Yau which lies inside the neighbourhood. This three-cycle has intersection 1 with the homology class of the which replaces the node. The homology classes of these two three-cycles will play an important role in the following.
Remember that the sphere can be given in the local coordinates around the deformed node by
Missing Equation
Also we can give the ball by
Missing Equation
The holomorphic three-form can be given locally by
Missing Equation
where the factor f is a holomorphic function which is finite in the whole neighbourhood. This determines the behaviour of the periods at small. For the periods over the vanishing three-sphere we get Missing Equation
where is the standard density on and the ellipses denote terms which are of smaller order in. The period over the three-cycle extending the ball can be approximated Missing Equation
where now the ellipses stand for terms which are analytic and of smaller order in. The boundary R is the radius of the neighbourhood. We assumed in this calculation that can be chosen in such a way that it meets no other nodes at. If this cannot be avoided, all nodes contribute the same divergence, as the local integral is unaffected. All the other periods are analytic and finite in. So we see that at the conifold point, which is at. the period over the vanishing vanishes as expected. Also it is important to note that the other periods, including the one over. remain finite. There is however a potential problem in the logarithmic behaviour of the latter period, which will show up later in derivatives of this period.
In the small resolution the of the base and the generator of the cone make up a four-ball. also indicated in figure 5.1, thus replacing the cone with. Now this is the cap of a four-cycle having intersection 1 with the blown up in.
One could think now that when going from to the second (co)homology changed by an amount of P for the P nodes, as there is an extra for every node. The 's corresponding to the different nodes may however be related in the homology of. The same is true for the 's in. Denote the number of these 's which have different homology classes by R. As the Euler number of a Calabi-Yau is given in terms of the Hodge numbers by. the change in the hodge numbers can generally be given by Missing Equation
The number R can also be seen in the homology of the deformation. There it is the number of homology relations between the 's replacing the different nodes. This can be seen as follows. We will denote the different 's by. Let there be a homology relation between these three-cycles, that is
Missing Equation
in the homology. This means that there is a four-manifold (often referred to as a four-chain) with as its boundary this combination
Missing Equation
When the vanish for. the boundary of the four manifold contracts to zero and hence becomes closed. This happens at the transition to the small resolution. Thus in the resolution there is an extra class in the fourth homology related to. When there are R independent homology relations between the then we find R extra homology classes in. and by PoincarŽ duality there must be also R new classes in. These are of course the homology classes of the 's replacing the nodes. The four-cycles are the cycles which extend the 's near the nodes.
A similar discussion could of course be held at the small resolution side of the conifold. There the should be the vanishing two-spheres while the relations between these are represented by a three-chain (three-manifold with boundary). After the conifold transition the relations become cohomology classes in the third cohomology of the deformation. Thus if there are S of these relations between the vanishing 's then we obtain S new classes in. This is in complete agreement with (5.51) if we take S=P-R.
A remarkable consequence of these different desingularizations is that it indicates that the moduli spaces of Calabi-Yau spaces which have completely different topology are connected to each other. In fact it seems that even all Calabi-Yau moduli spaces can be connected by processes similar to the one we described here. To really show that they are connected one should show that the conifold singularity is at finite distance in the moduli space. This can be done quite explicitly by considering the periods of. The only potential infiniteness comes from the periods over the three-cycles and in the neighbourhood of the node. So let us choose a symplectic basis for such that is the vanishing sphere and is the three-cycle having intersection 1 with. Then for small the period vanishes as by (5.49), hence the conifold point is at. Using then (5.50) we find asymptotically for as a function of
Missing Equation
where we only kept the lowest order terms which are not analytic at. The only potential singularity in the metric now comes from the component. This component near behaves like. so has as its most singular part a divergence. As any line integral of this function near is finite, this shows that the singularity is at finite distance. The metric however develops a curvature singularity at the conifold point, as can be verified directly from the result above.
In the previous section we have encountered a serious breakdown of the effective low energy description of the type IIB string near a conifold singularity. This singularity is logarithmic, and really of precisely the same nature as the strong coupling singularity we encountered in the N=2 super Yang-Mills theory. There we argued that the natural explanation of such singularities is that states which are massive at generic points in the moduli space become massless at the singular points. In the rigid case these massless points were BPS saturated monopoles and dyons. Recently it has been proposed by Strominger in [50] that in type IIB string theory there are charged black hole states that become massless at the conifold points. It should be no big surprise that these objects are black holes, as gravity is included in our string theory.
We will now discuss how these black holes arise in string theory and how they are related to the vanishing three-cycles. Remember that the singularities we discussed arise in type IIB theory. We begin by discussing the type IIB theory in 10 dimensions, and after this discussion compactify on the Calabi-Yau. The vector multiplets in this theory are derived from the 10-dimensional self-dual 5-form field strength, which we shall here denote simply by. The conserved charges associated to this field are given by
Missing Equation
for any five-cycle (remember that is self-dual). These charge should be carried by a 3+1 dimensional object, or a three-brane, as such a space can be enclosed by a five-cycle in 10 dimensions. There are three-brane solitonic solutions to the field equations of the low energy effective action of the type IIB theory. (They have a three-brane structure because they have 3+1 dimensional translational symmetry). It turns out that the dilaton can be set to a constant in the field equations as it does not couple directly to the 5-form. The (classical) field equations for the self-dual five-form field strength and the metric are then given by Missing Equation
The three-brane solution to this equation is given by (see [52])
Missing Equation
where Q is the charge with respect to any five-cycle enclosing the three-brane, is the density on the transverse five-sphere normalized such that and. are coordinates along the five-brane and the function f(r) is a fundamental solution to the Laplace equation on the transverse 5-space
Missing Equation
This is not a general solution, but it is the solution with the lowest mass and the smallest horizon, and the only solution which does not radiate Hawking radiation and can therefore be stable. Also this solution preserves one of the supersymmetries of the type IIB supergravity. It can be shown to provide a good background for the string theory, i.e. it can be promoted to all orders in to a background for a superconformal field theory. It is a regular solitonic solution with a horizon at. For example the scalar curvature is, using the field equations (5.66),
Missing Equation
which is perfectly regular near at horizon.
Now we compactify the string to 4 dimensions on a Calabi-Yau manifold M. It can be expected that we can generalize the black three-brane solution above to this curved space time. It can then wrap around three-cycles on the Calabi-Yau manifold. In the four dimensional effective theory these solutions appear as ordinary localized black holes. The charges of this black hole with respect to the dimensionally reduced gauge fields are given by
Missing Equation
where is a two-sphere in the four dimensional space time enclosing the black hole. These are the electric and magnetic charges of the dimensionally reduced gauge fields. Quantization of charges restricts them to integers, when properly normalized. Because the solution is supersymmetric it must satisfy a BPS bound. This bound is here given by
Missing Equation
The black holes form a complete multiplet of states because of the supersymmetry transformations that act on them. It can be shown that the black holes we discussed are members of hypermultiplets. We know that when a hypermultiplet which is integrated becomes light the coupling develops a singularity
Missing Equation
where M is the (small) mass of the hypermultiplet. Especially, consider the black hole hypermultiplet with as its only vanishing charge. This multiplet has mass. At the vanishing locus of the corresponding three-cycle this mass becomes zero. Then the coupling of the first gauge field develops a singularity
Missing Equation
We will now see that such singularities can be used to explain the singularities that arise at the conifold points in the moduli space.
Let us denote the black hole multiplet corresponding to the vanishing cycle by. The gauge field of this solution can be written as the self dual part of. where denotes the three-form dual to. and is the fundamental solution of an ordinary field strength. Therefore the charges of this hypermultiplet are given by
Missing Equation
while the other charges vanish. Hence the mass of the hypermultiplet is given by
Missing Equation
which vanishes precisely at the vanishing locus of the corresponding three-cycle. Assume now that these are the only states which become massless at the singular locus located at. We then obtain a singularity in the gauge coupling near the singular locus
Missing Equation
With the relation of the couplings to the special geometry we find from this a singularity in the dual coordinates
Missing Equation
This is precisely the form of the singularity we found for this function in equation (5.64). This shows that the inclusion of the black hole hypermultiplets in the effective action cures the singularity at the conifold points.
We have seen now what the physical implications are of the vanishing cycles in terms of massless black holes. The homology relations between the vanishing three-cycles also have physical significance. To see how this comes about we study the scalar potential of the hypermultiplets. Let us denote by h the scalars of a hypermultiplet H. This field is a complex vector in the fundamental representation of. Denote the n dimensional complex irreducible representation of by. so that h takes values in. Then the antisymmetric tensor product takes its values in. The charges of the hypermultiplet H form an element of. the fibre of the symplectic bundle of special geometry (or rather its dual). Denote this vector by Q. Then the scalar potential is given by
Missing Equation
where we used the natural inner products on and. This applies to a single hypermultiplet. In our discussion we consider P hypermultiplets. We can write the scalar potential as Missing Equation
where
Missing Equation
which takes values in.
There are some possible flat directions which this potential admit. We can choose all the vacuum expectation values of the scalars equal to zero. We can then choose nonzero values for the scalars in the vector multiplets. This gives masses to the black hole hypermultiplets. Physically this is the Higgs mechanism and it takes us back to the Calabi-Yau phase of the deformation, in which the vanishing three-cycles have nonzero volume.
When there are homology relations between the vanishing cycles we can find more flat directions. Remember that these relations are linear relations between the
Missing Equation
As the charges are coordinates of the in the homology, this implies analogous relations between these charges. Therefore if we choose the vector proportional to a relation. or more explicitly Missing Equation
where h is an element of independent of a, then the potential vanishes. Choices of this kind are the only flat directions of this potential, and different choices of directions for h are related by gauge transformations. Therefore we find in the case of R homology relations between the vanishing cycles also R flat directions in the potential for the hypermultiplets. Moving along these flat directions generically gives masses to the gauge fields under which the P hypermultiplets are charged. There are P-R of these gauge fields, because they are related to the homology classes of the. To obtain massive multiplets these vector multiplets must pair up with as many hypermultiplets. Therefore for generic large values of the scalars. P-R vector multiplets and P-R hypermultiplets disappear from the low energy spectrum. This implies that generically the number of massless hypermultiplets is increased by an amount of R instead of the P black hole hypermultiplets. This then brings us smoothly to a new phase in the physical moduli space. We could wonder if this phase can also be represented by a Calabi-Yau compactification. If this is the case we should find a Calabi-Yau M' which satisfies Missing Equation
This is exactly the relation we found between the small resolution and the deformation in (5.51). Therefore we are led to choose M' equal to a small resolution. Remark that this does not completely fix M' as there are several topologically distinct resolutions. Detailed analysis of the moduli spaces should give more informations. In [53] a more detailed analysis using the techniques of toric geometry are used to find the right resolution. This can also be used to proof this identification. This will however lead us too far, so we will not go further into it. A transition to the Calabi-Yau phase of the resolution implies that we have to identify the expectation values of the modulo gauge transformations with the volumes of the R independent cycles replacing the nodes. So we see a direct relation between the geometry and the black hole states. In the transition the non-perturbative black hole states in the deformation phase go over into ordinary perturbative states in the small resolution phase. Thus there is no real distinction between the black hole states and perturbative elementary particle.
As a final remark, let us go back to the simple case where we have no relation between the cycles, that is R=0. Then the above analysis seems to give the wrong answer. This follows from the fact that there are no flat directions in the potential (5.82) in the direction of the hypermultiplet moduli space because we have no choice like (5.85) available. Hence the related conifold transition described above does not exist in this case. The formula (5.86) suggests however that it does exists and also geometrically we can construct the conifold transition as we have seen earlier. The formula (5.86) suggests that the number of vectors will jump (remember that the number of vector moduli is ). This does however not happen as we lack the required flat direction. We rather expect the number of hypermultiplets to jump, contrary to what (5.86) suggests. How can this be reconciled with the geometrical transition? The answer, as was recently shown in [54], is that the resolution of the Calabi-Yau when R=0 is not KŠhler and hence it can not (directly) be used for type II compactifications. The reason for this is that for any vanishing three-cycle there exists a three-cycle which has intersection 1 with this vanishing cycle and no intersection with any of the other vanishing cycles. After the transition to the resolution these (non-vanishing) cycles open up and become three-chains. The boundaries are the blown-up two-spheres. Because any of these spheres is the boundary of a three-chain, they are all homologically trivial. Homologically trivial two-cycles however can not exist on a KŠhler manifold, because for any. Hence the resolution is not KŠhler whenever two of the vanishing three-cycles are not contained in a homology relation. Therefore the simple case we studied at first has some problems when applied to the global moduli space. The analysis is therefore only valid locally near the singularity.
We can do much of the analysis above as well for the type IIA theory near a conifold point. The vector moduli space of this theory is given by the 'quantized' KŠhler moduli space of the Calabi-Yau manifold. Therefore the conifold singularity will turn up in the moduli space of the small resolution of the conifold. The singularity turns up in the point
Missing Equation
where is the vanishing two-cycle. This point lies at the boundary of the KŠhler moduli space as the KŠhler form is degenerate here. The prepotential of type IIA is given in (5.32). If there is only one vanishing cycle the singularity is given by the corresponding contribution to the instanton sum
Missing Equation
This singularity is of precisely the same nature as the singularity we encountered in the type IIB moduli space.
We can now try to find a solitonic solution of type IIA theory with vanishing mass at the conifold point. This soliton should be related to the vanishing two-cycle and therefore is expected to be a two-brane. Such an object can be charged with respect to the four-form field strength of the 10-dimensional type IIA theory. The classical field equations for this field strength following from the action (5.15) are given by
Missing Equation
These equations have an electrically charged two-brane solution given by (see [52])
Missing Equation
where is the density on the transverse 6-sphere,. i=8,9, are the coordinates along the two-brane and the function f(r) is a fundamental solution to the Laplacian on the transverse 7-space
Missing Equation
This solution is however singular at the horizon located at. This can be seen from the field equations (5.89) in combination with the solution, which gives a scalar curvature
Missing Equation
which is singular at the horizon. Therefore this solution is classically not a regular solitonic solution. Quantum corrections could however cure the singularity. The analysis of these quantum corrections is however much to complicated and not enough is known about the quantum gravity near horizons to say much about this. It is however not surprising that we can not cure the singularity at the lowest order level. At the lowest order we did not even find a singularity. This can be seen from the prepotential in (5.32). Perturbatively in the prepotential of type IIA is given only by the cubic intersection piece and possibly some lower order contributions. These are completely regular globally in the moduli space, including the conifold points at the boundary. The singular contributions only are found in the nonperturbative (in ) instanton contributions. In the effective field theory these correspond to nonlocal corrections to the effective Lagrangian (remember that the perturbative corrections correspond to higher derivative terms). Therefore we could never correct these terms perturbatively using only the local effective Lagrangian.
We do however expect that after all these corrections are taken into account the solutions given above reduce to a perfectly regular soliton in string theory. There is another reason to believe this. It is well known that the type IIA supergravity Lagrangian is the dimensional reduction of the 11 dimensional supergravity Lagrangian compactified on a circle. The latter theory admits a two-brane solitonic solution, which is perfectly regular. The solution in 10 dimensions above can be seen as the dimensional reduction of this solution. Therefore if 11 dimensional supergravity corresponds to a well behaved quantum theory of which string theory is just a limit, the solution above is then just a limit of some regular solution in this theory. This 11 dimensional theory is conjectured to exists and is known as M-theory in the literature. It is however not been achieved as a well defined quantum theory like string theory. It cannot be a string theory because strings are not allowed in 11 dimensions. The fundamental objects of M-theory are conjectured to be membranes. They do exist as solitonic solutions in the 11 dimensional supergravity, which should be its effective field theory. Recently much progress has been made with this theory, although much is yet unsolved.
So for now let us assume that these solitonic objects exist in the 10 dimensional type IIA theory. When we compactify this theory on a Calabi-Yau manifold we still expect a related soliton to exist. When we wind the two-brane around a two-cycle in the Calabi-Yau then we again find a black hole state in the four dimensional effective field theory. The analysis we had for the type IIB theory now goes through much in the same way. So we obtain a complete hypermultiplet of states for each vanishing two-cycle which has a BPS mass
Missing Equation
which goes to zero at the conifold locus. We find monodromies from the instanton corrections. Furthermore when we have P nodes on the conifold with in this situation S relations between the vanishing two-cycles, we obtain in the same way as for type IIB a phase transition at the conifold point. There the P black holes have vanishing masses. The scalar potential of these hypermultiplets is given by effectively the same formula as before, and therefore has S flat directions. The black holes can again condensate along these flat directions, where generically P-S vector multiplets pair up with as many charged hypermultiplets (from the black holes) and become massive. Hence generically this transition has S extra hypermultiplet states and P-S less vector multiplets. Remember now that vector multiplets in type IIA are related to and hypermultiplets to. Therefore if the transition brings us to a Calabi-Yau phase with Calabi-Yau space M' then it induces a change in the Hodge numbers
We have seen that there are roughly three different types of (closed) superstring theories. They all have their own characteristics and when compactified to four dimensions they all give rise to N=2 supergravity. We have also seen that in the context of super Yang-Mills different N=2 field theories can be smoothly connected in the moduli space. E.g. the moduli space of pure N=2 super Yang-Mills with gauge group is connected to a theory coupled to a charged hypermultiplet (the monopole or the dyon). We can wonder if we can do similar things if we include gravity, that is for supergravity theories. Using the results we have represented so far we can already give some clues for the string theories, which are an extension to supergravity, which indicate such a duality. We have seen already how a transition to a completely different string theory can arise through an analysis of the conifold points. These transitions connected a string theory of type IIB (or type IIA) with a string theory of the same type. We can wonder if we can do more and connect different types of strings. This would give us a really non-perturbative extended description of string theory. Seiberg-Witten theory already indicates this type of duality. The Yang-Mills theory is naturally embedded in a heterotic string, as this is the only (closed) string theory which can contain nonabelian gauge fields at a perturbative level. So the weak coupling limit in super Yang-Mills is naturally described by a heterotic string when we include gravity. By duality and the analysis of super Yang-Mills we should expect to find extra singularities in the moduli space where non-perturbatively generated charged matter becomes massless. This would also be happening in the string theory as this is an extension to Yang-Mills. We saw such a thing happening most natural in the context of type II strings. We can therefore wonder if the different string theories describe just different limits of the same theory. This would be a true unification of string theory. In the last few years much hints have been given for these kinds of duality. We will here describe how such dualities can arise and how the calculations we did support these conjectured duality.
We will begin by a duality called mirror symmetry for which quite some evidence is gathered now. An introduction to the subject and some can be found in [55], while some more detailed calculations are performed e.g. in [56] and [57]. To see this first observe that exchanging and in the superconformal algebra and at the same time replacing the current J by -J defines an automorphism of this algebra. For type II theory on a Calabi-Yau the internal sector is described by a (2,2) superconformal field theory. For such a model we can apply this symmetry transformation to the left moving sector only. This should then give an equivalent superconformal model. This resulting model is however not the same in general. The chiral ring and the anti-chiral ring are exchanged by this symmetry. This equivalence of conformal theories is called mirror symmetry.
If we now define the superconformal theory as a -model on a Calabi-Yau manifold M we can apply mirror symmetry to this model. It can then happen that the resulting superconformal model can again be described by a -model on a (different) Calabi-Yau manifold. This Calabi-Yau is called the mirror manifold of M. This manifold has a topology which is generally very different from that of the original Calabi-Yau M. This can already be seen from the Hodge numbers of the two manifolds. The hodge numbers determine the ranks of the (anti) chiral rings. We have seen that the rank the (a,c) ring is related to the KŠhler deformations while the (c,c) ring is related to the complex structure deformations. Hence mirror symmetry exchanges the hodge numbers and. Remark that this means that it inverts the Euler number of the manifold. This induces a reflection in the Hodge-diamond of the Calabi-Yau. This is the origin of the name mirror symmetry for this equivalence. There are now found many examples of mirror pairs. A complete proof of mirror symmetry in the context of Calabi-Yau manifolds is still lacking, but much has already been checked. Calabi-Yau manifolds can quite generally be described using the techniques of toric geometry. Using this description a quite beautiful construction has been found to find the mirror of a Calabi-Yau manifold. Really mirror symmetry is a form of T-duality. Indeed, in the context of torus compactifications, which can also be Calabi-Yau manifolds, the exchange of complex structure moduli and KŠhler moduli is just a T-duality transformation. Also in the context of general Calabi-Yau manifolds the exchange symmetry is just a perturbative symmetry from the space-time point of view, just as T-duality transformations. In fact mirror symmetry is only a duality on the level of the CFT, i.e. the classical background of the string. This duality should even be true quantum mechanically for the type II string, interchanging the two types of string [58][59][60]. This would make possible many new non-perturbative calculations in string theory.
Another conjectured duality is even more challenging. It is the duality of the heterotic string to the type II string (remark that with the mirror symmetry conjecture we have really one type II theory). That such a duality might arise already was mentioned above comparing it to the non-perturbative duality in Yang-Mills theory. This duality was first proposed in [62]. Some evidence has been gathered in favor of this duality. Especially in N=4 compactifications of these theories this duality is already quite established. There a solitonic solution to the type IIA action has been found which has precisely the right massless spectrum to identify it with the heterotic string. A great advantage of the N=4 theory is that there are no quantum corrections to the theory as the -functions vanish in this case. In the N=2 case it is much harder to proof this duality, as it really would involve both perturbative and non-perturbative physics, and we do not have a good way to calculate with non-perturbative physics. We will now see how this conjectured duality can arise and what it implies for the calculations we did.
To find a duality between the heterotic string and the type IIA string we should compare the prepotential of these theories. To make this comparison it is at first necessary to translate the KŠhler moduli of the type IIA string to the moduli of the heterotic string. This comparison can best be done by comparing tree level heterotic string result to the (worldsheet) perturbative result for the type IIA string. These are both third order in the moduli. An important consequence of this is that for the heterotic string these are first order in the dilaton S, while all the other perturbative corrections are independent of S. This dilaton should be identified with a KŠhler modulus on the type IIA side. We call the corresponding basis element in . On the type IIA side the cubic terms are determined by the intersection matrix of the corresponding two-forms. Therefore the KŠhler modulus should satisfy
Missing Equation
for any basis element. Hence we should have (in cohomology). Furthermore it should satisfy a positivity condition due to its relation to the couplings, cf. [63]. The existence of such a two-form puts some constraints on the possible Calabi-Yau on which the type IIA is compactified. It is known that such a Calabi-Yau must be a K3-fibration [63]. This means that the Calabi-Yau is a fibred space with base and generic fibre a K3 manifold. The direction corresponding to the dilaton is by PoincarŽ duality related to the class of the basis. Remark that the small dilaton limit, which is the perturbative limit of the heterotic string, is related to a small fibre, or a large base. In this limit we expect the fibre structure to become less important and therefore the Calabi-Yau to have the structure of. This is then the manifold on which the heterotic string is compactified.
After the identification of the dilaton we can compare the perturbative heterotic results to the results of the dual type IIA string. Comparing the results of the prepotential in (4.95) and (5.32) we see directly a completely similar expansion on the two sides. Only the coefficients are determined by different geometrical properties. On the heterotic side they are related to global structures of the gauge bundle, while on the type IIA side they are related to counting of curves. If we have a dual pair we can compare these coefficients. The coefficients on the type IIA side which are not related to the dilaton. that is for curves which do not intersect the base of the fibration (which is in the class of ), can directly be identified with coefficients c(n) in the perturbative heterotic result. The other contributions from curves intersecting should then be identified with non-perturbative contributions on the heterotic side. As the type IIA result is complete, this would give the full non-perturbative results for the heterotic string. This is certainly a wonderful result. Using this duality already several predictions have been made for the full non-perturbative couplings in heterotic string theory, see for example in [61], [62] and [64].
Especially we can compare the three moduli case as we have seen this for the heterotic side in some more detail. The perturbative corrections involve a cubic part which is given by. We argued there that this result was quite general. In [64] the moduli S, T and U were identified with KŠhler moduli for a number of known Calabi-Yau manifolds. In all these cases they found, after this identification, the same cubic terms on the type IIA side.
Apart from the prepotential, which governs the couplings between vector multiplets, there are further couplings which can be studied in string theory. These should correspond to higher derivative terms in the effective action. Especially the coupling between gravitons and vector multiplets has attracted much attention recently. These are described by a whole range of interactions of the form
Missing Equation
where W describes (the anti self-dual part of) the graviton multiplet (including the graviphoton and the graviton), X denotes the vector multiplets. We see that the function describes the graviton coupling we mentioned before, after (4.35). The higher order corrections can be calculated in the various string models. For type II strings they can be calculated using a recursion relation which can be found using direct worldsheet calculations. This has been done in [65] and [66]. It turns out that there the correction corresponds to the supersymmetric partition function at genus g. The same couplings can also be calculated in heterotic string theory. The same reasoning as for the prepotential restricts these calculations here to genus 0 and genus 1. Only and turn out to have tree-level contributions. In this setting they can be calculated using the same techniques as were used to calculate the threshold corrections [67]. It turns out, as was shown in the last reference, that these contributions from the two string theories agree, giving more support for the heterotic-type II duality. These corrections were also used in [68] in the neighbourhood of a conifold to give support for the idea of Strominger for the resolution of this singularity.
A special feature of this duality is that it interchanges target space perturbative and even non-perturbative calculations from the heterotic side with worldsheet perturbative and non-perturbative calculations on the type IIA side. The latter are often easier to handle as they are related to calculations in a 2-dimensional conformal field theory. Furthermore the type IIA result can be calculated exactly using mirror symmetry. And these results are even obtained by a classical calculation with respect to any perturbation. Therefore a complete non-perturbative treatment becomes possible using such dualities. In certain cases this gives also some new dualities on the heterotic side, by dualizing twice in different ways to a type II string [69].
For this heterotic-type II duality to work, we need a whole new range of non-perturbative physics. Perhaps the most striking new physics is the enhancement of symmetry. As we have seen the heterotic string allows non-abelian gauge symmetries because it has a bosonic string sector, hence allowing a Frenkel-Kac mechanism. As the type II string is dual to the heterotic string, also here we should expect enhanced symmetries at certain points in the moduli space. These can however not arise at the perturbative level because the Frenkel-Kac mechanism does not work in these models. Therefore these gauge bosons should be non-perturbative in nature. This is also what should be expected when one calculates the masses of these states as expected from duality. It turns out that these enhanced symmetries arise precisely at points in the moduli space where the Calabi-Yau space is singular [70]. Therefore we should expect a mechanism similar to the one we used for the resolution of the conifold to apply here [71][72].
All this non-perturbative behaviour in the various string models should in some way be described by non-perturbative states. These states should describe the hypermultiplets resolving the conifolds, the enhanced symmetries in type II strings and other states required by duality. These can not be found directly in the CFT, as this can only give the genus expansion, which is equivalent to the perturbative expansion in the string coupling constant. But recently a whole new field opened up when these non-perturbative states were identified with so called Dirichlet branes or D-branes [73]. These objects are submanifolds, or p-branes, on which a (closed) string can have endpoints (an introduction to the subject can be found in [74]). They have a similar behaviour to instantons, but they can be described using conformal field theory (but now on a surface with boundaries). These states already passed many tests on properties predicted by duality. So it is generally believed within the field that they are the states required for duality. This really gives a way to give a perturbative description of non-perturbative physics. But the description is yet far from complete, and much has still to be done to have a complete description of a full, non-perturbatively described, theory.