.

RUNHETC-2003-36

hep-th/0401049

Les Houches Lectures on Strings and Arithmetic

Gregory W. Moore^{†}^{†} Summary of
lectures delivered at the conference Number Theory,
Physics, and Geometry Les Houches, March, 2003

Department of Physics, Rutgers University

Piscataway, NJ 08854-8019, USA

These are lecturenotes for two lectures delivered at the Les Houches workshop on Number Theory, Physics, and Geometry, March 2003. They review two examples of interesting interactions between number theory and string compactification, and raise some new questions and issues in the context of those examples. The first example concerns the role of the Rademacher expansion of coefficients of modular forms in the AdS/CFT correspondence. The second example concerns the role of the “attractor mechanism” of supergravity in selecting certain arithmetic Calabi-Yau’s as distinguished compactifications.

Jan. 6, 2004

1. Introduction

Several of the most interesting developments of modern string theory use some of the mathematical tools of modern number theory. One striking example of this is the importance of arithmetic groups in the theory of duality symmetries. Another example, somewhat related, is the occurance of automorphic forms for arithmetic groups in low energy effective supergravities. These examples are quite well-known.

In the following two lectures we explore two other less-well-known examples of curious roles of number theoretic techniques in string theory. The first concerns a technique of analytic number theory and its role in the AdS/CFT correspondence. The second is related to the “attractor equations.” These are equations on Hodge structures of Calabi-Yau manifolds and have arisen in a number of contexts connected with string compactification. Another topic of possible interest to readers of this volume will appear elsewhere [1].

2. Potential Applications of the AdS/CFT Correspondence to Arithmetic

2.1. Summary

In this talk we are going to indicate how the “AdS/CFT correspondence” of string theory might have some interesting relations to analytic number theory. The main part of the talk reviews work done with R. Dijkgraaf, J. Maldacena, and E. Verlinde which appeared in [2]. Ideas similar in spirit, but, so far as I know, different in detail have appeared in [3].

2.2. Summary of the AdS/CFT correspondence

The standard reviews on the AdS/CFT correspondence are [4,5,6]. In this literature, “anti-deSitter space” comes in two signatures. The Euclidean version is simply hyperbolic space:

while the Lorentzian version is

where on the right-hand side we should take the universal cover. These spacetimes are nice solutions to Einstein’s equations with negative cosmological constant.

In the context of string theory they arise very naturally in certain solutions to 10- and 11-dimensional supergravity associated with configurations of branes.

Some important examples (by no means all) of such solutions include 1. where is a Calabi-Yau -fold. The associated D-brane configurations are discussed in Lecture II below.

2. where is a surface or a torus , or .

3. . This is the geometry associated to a large collection of coincident branes in 10-dimensional Minkowski space and is the subject of much of the research done in AdS/CFT duality.

At the level of slogans the AdS/CFT conjecture states that 10-dimensional string theory on

is ‘‘equivalent’’ to a super-conformal field theory -- i.e., a QFT without gravity -- on the conformal boundary

The “conformal boundary of AdS” means, operationally,

More fundamentally it is the conformal boundary in the sense of Penrose.

Of course, the above slogan is extremely vague. One goal of this talk is to give an example where the statement can be made mathematically quite precise. We are explaining this example in the present volume because it involves some interesting analytic number theory. The hope is that a precise version of the AdS/CFT principle can eventually be turned into a useful tool in number theory, and the present example is adduced as evidence for this hope. At the end of the talk we will make some more speculative suggestions along these lines.

2.2.1.AdS/CFT made a little more precise

In order to explain our example it is necessary to make the statement of AdS/CFT a little more precise.

Consider 10D string theory on which is a noncompact manifold which at infinity looks locally like

Let us think of string theory as an infinite-component field theory on this spacetime. In particular the fields include the graviton , as well as (infinitely) many others. Let us denote the generic field by . We assume there is a well-defined notion of a partition function of string theory associated to this background. Schematically, it should be some kind of functional integral:

Even at this schematic level we can see one crucial aspect of the functional integral: we must specify the boundary conditions of the fields at infinity.

Since spacetime has a factor which is locally at infinity there is a second order pole in the metric at infinity. Let denote a coordinate so that the conformal boundary is at and such that the metric takes the asymptotic form

where denote coordinates on . In these coordinates we impose boundary conditions on the remaining fields:

The functional integral (2.8) is thus a function
^{†}^{†} In fact, it should be considered as a “wavefunction.” In the closely
related Chern-Simons gauge theory/RCFT duality this is literally true. of the boundary data:

We can now state slightly more precise versions of AdS/CFT. There is a slightly different formulation for Euclidean and Lorentzian signature.

The Euclidean version of AdS/CFT states that there exists a CFT defined on such that the space of local operators in is dual to the string theory boundary conditions:

such that

This statement of the AdS/CFT correspondence, while conceptually simple, is quite oversimplified. Both sides of the equation are infinite, must be regularized, etc. See the above cited reviews for a somewhat more careful discussion.

The Lorentzian version of AdS/CFT states that there is an isomorphism of Hilbert spaces between the gravity and CFT formulations that preserves certain operator algebras. These are , the Hilbert space of the CFT on , and , the Hilbert space of string (or M) theory on . This is already a nontrivial statement when one considers both sides as representations of the superconformal group. An approximation to is given by particles in the supergravity approximation, and corresponding states in the CFT have been found. See [4]. Whether or not the isomorphism truly holds for the entire Hilbert space is problematic because of multi-particle states and because of the role of black holes. Indeed, it is clear that one must include quantum states in associated both to black holes and to strings and D-branes in order to avoid contradictions.

2.3. A particular example

In the remainder of this talk we will focus on the example of type string theory on . In this case the dual CFT on is a two-dimensional CFT .

From symmetry considerations it is clear that the dual CFT has supersymmetry. It is thought that admits marginal deformations to a supersymmetric -model whose target space is a hyperkahler resolution

In comparing the gravity and CFT side we make the identification

where is the radius of (which in turn is the curvature radius of ), while is the Newton constant in 3 dimensions. The quantization of can be seen intrinsically on the gravity side from the existence of certain Chern-Simons couplings for gauge fields with coefficient .

The “proof” of the correspondence proceeds by studying the near horizon geometry of solutions of the supergravity equations representing D1 branes and D5 branes wrapping . One studies the low energy excitations of the “string” wrapping the factor. The dynamics of these excitations are are described by a supersymmetric nonlinear sigma model with target space (2.14) for . The moduli space of supergravity solutions, as well as the moduli space of the supersymmetric sigma model are both the space

where is an arithmetic subgroup of . See [7,8,9,10,11] for some explanation of the details of this.

The correlation function whose equivalence in AdS and CFT formulations we wish to present is a certain parititon function which, on the CFT side is the elliptic genus of the conformal field theory. The reason we focus on this quantity is that the dual CFT is very subtle. The elliptic genus is a “correlation function” of the CFT which is invariant under many perturbations of the CFT, and is therefore robust and computable. Nevertheless, the resulting function is also still nontrivial and contains much useful information.

Our strategy will be to write the elliptic genus in a form that makes the connection to quantum gravity on clear. The form in which we can make this connection is a Poincaré series for the elliptic genus.

2.4. Review of Elliptic Genera

For some background on the elliptic genus, see [12,13,14,15,16,17,18,19,20,21,22].

Let be a CFT with supersymmetry. This means the Hilbert space is a representation of superconformal , where the subscript refers to the usual separation of conformal fields into left- and right-moving components.

Let us recall that the superconformal algebra is generated by Virasoro operators , and current algebra , with , and superconformal generators with for the NS algebra and for the R algebra. The commutation relations are:

Right-moving generators are denoted .

The elliptic genus is

where the trace is in the Ramond-Ramond sector and .

In a unitary superconformal field theory the operators may be simultaneously diagonalized. In a unitary theory the spectrum satisfies in the Ramond sector (and similarly for the right-movers). States with are called right-BPS. It follows straightforwardly from the commutation relations (2.18) that only right-BPS states make a nonzero contribution to the trace (2.23) and hence has Fourier expansion

where and .

In this paper we will be considering superconformal theories with supersymmetry. These are special cases of the theories, but have extra structure: For each chirality, left and right, the current algebra (2.19) is enhanced to a level affine current algebra . In addition, for each chirality, there is a global symmetry and the four supercharges transform in the . The Virasoro central charge is given by . representation of the global

2.4.1.Properties of the Elliptic Genus

The elliptic genus satisfies two key properties: modular invariance and spectral flow invariance. The modular invariance follows from the fact that can be regarded as a path integral of on a two-dimensional torus with odd spin structure for the fermions.

Under modular transformations

In order to prove this from the path integral viewpoint note that including the parameter involves adding a term to the worldsheet action. From the singular ope of with itself one needs to include a subtraction term. After making a modular transformation this subtraction term must change, the difference is finite and accounts for the exponential prefactor in (2.25).

The algebra has a well-known spectral flow isomorphism [23]

which implies that

The identities (2.25) and (2.27) above are summarized in the mathematical definition [24]:

Definition A weak Jacobi form of weight and index satisfies the identities:

and has a Fourier expansion with unless .

Thus, the elliptic genus of a unitary superconformal field theory is a weak Jacobi form of weight and level . Much useful information on Jacobi forms can be found in [24].

Using the spectral flow identity (2.29) we find that , for an integer, and therefore if . Using this it is straightforward to derive

Here are level theta functions

We denote the combinations even and odd in by .

Our goal now is to write the elliptic genus for the conformal field theory appearing in the AdS/CFT correspondence in a fashion suitable for interpretation via AdS/CFT. This fashion will simply be a Poincaré series. Before doing this in section 2.5 we make a small digression.

2.4.2. Digression 1: Elliptic Genera for Symmetric Products

If the conformal field theory is a sigma model with target space , denoted , then the elliptic genus of the conformal field theory only depends on the topology of and hence we can speak of In this case can be interpreted as an equivariant index of the Dirac operator on the loop space . The parameter accounts for rigid rotations of a loop, while accounts for rotations in the holomorphic tangent space of the target.

We will be considering the elliptic genus for the case . The elliptic genus for such is expressed in terms of the elliptic genus of itself. For any conformal field theory with Hilbert space we can consider the symmetric group orbifold of . Denote the Hilbert space of the orbifold theory by . This has a decomposition into twisted sectors given by

where the sum is over partitions of :

The space is isomorphic to the space . It corresponds to “strings of length ” where we scale the usual parameter by a factor of . Thus configurations in the symmetric product orbifold theory may be visualized as in Fig. 1.

Fig. 1: A configuration of strings in the symmetric product conformal field theory.

Now, if is a conformal field theory based on a sigma model with target space then (2.32) implies an identity on the orbifold elliptic genus for . To be specific, if

then [25]

In the AdS/CFT correspondence we apply this to . The elliptic genus of can be computed (say, from orbifold limits or Gepner models) and is

and therefore, is explicitly known.

The decomposition in terms of theta functions is given by [26]

with

with

Many other interesting aspects of the elliptic genus of and its symmetric products, including relations to automorphic infinite products can be found in [27].

2.5. Expressing the elliptic genus as a Poincaré Series

Returning to our main theme, we will explain the basic formula first in a simplified situation. Then we state without proof the analogous result for weak Jacobi forms. The proof may be found in [2].

Let be a weak modular form for of weight . The adjective “weak” means that is allowed to have a pole of finite order at the cusp at infinity, but no other singularities in the upper half plane. Thus, the Fourier expansion of takes the form:

We refer to the finite sum

as the polar part.

In the physical context, , for a unitary CFT, where is the central charge of the Virasoro algebra. Moreover, , where is the number of noncompact bosons in the CFT. Unfortunately, the letters are quite standard in the theory of modular forms so there is a clash of conventional notations. We will try to avoid the use of for central charge and noncompact dimensions in what follows and use instead.

It turns out to be essential to introduce a map

The explicit map is

The fact that the right hand side of (2.44) is a modular form is sometimes called Bol’s identity. Note that in terms of the Fourier expansion we have:

where

Given a polynomial in one can construct by hand a modular form of weight by averaging over the modular group to produce a Poincaré series

Note that we must sum over cosets of the stabilizer of , that is, we sum over where

The resulting sum is convergent for .

In general, weak modular forms of positive weight are not uniquely determined by their polar parts. If the space of modular forms is nonzero one can always add an nonzero element to (2.47) to produce another form with the same polar part. However, if a form is in the image of the map (2.44) then it is in fact completely determined by its polar part. To see this, first note that has no constant term. Next we use a pairing between weak modular forms and cusp forms which was quite useful in [28]. If and is a cusp form then we can extend the Petersson inner product by

Here is the intersection of the standard fundamental domain of with the set of with . Using integration by parts we can see that is orthogonal to the space of cusp forms , and hence it is determined by its polar part.

Let us summarize: We can reconstruct from the polar part

(which is a finite sum) via

This is the kind of formula we are going to interpret in terms of AdS/CFT.

2.5.1.Digression 2: Rademacher’s formula

In the next two subsections we pause to make two more small digressions concerning some related issues: Rademacher’s formula, Cardy’s formula, and the applications to black hole entropy.

The Rademacher formula is a formula for the Fourier coefficients of which is particularly useful for questions about the asymptotic nature of the Fourier coefficients. The formula is easily derived from (2.51) by taking a Fourier transform. On the left hand side we have:

on the right hand side, after a little manipulation we have a sum of integrals of the form:

which can be expressed in terms of Bessel functions. The precise relation we find is

where is the Bessel function growing exponentially at

while

is a Kloosterman sum.

While (2.54) is a terribly complicated formula, it is in fact also very useful since it gives the asymptotics of Fourier coefficients of modular forms for large . In fact, it can be a very efficient way to compute the Fourier coefficients exactly if they are known, for example, to be integral.

In the physics literature the leading term,

is known as “Cardy’s formula.” It gives the “entropy of states at level ”

The subleading exponential corrections are organized in a beautiful way by Farey sequences. See [29,30,31] or [2], appendix B for details.

2.5.2.Digression 3: Black hole entropy

One very striking application of Cardy’s formula in the string literature is to the statistical accounting for the entropy of certain special black holes. This was first proposed in a famous paper of A. Strominger and C. Vafa [32].

As we have mentioned, the spacetime is obtained as a near-horizon geometry from a limit of a system of -branes and -branes wrapping . The “BPS states” of this system of branes correspond to special black hole solutions of 5-dimensional supergravity. The black hole solution is characterized by three charges . In the D-brane system, specify quantum numbers of BPS states; there is a -graded vector space of such states: , with charges . The elliptic genus counts the super-dimension of these vector spaces of BPS states:

The Cardy formula then gives:

and confirms the supergravity computation of the Beckenstein-Hawking entropy [32].
^{†}^{†} It is important to bear in mind that this is actually counting with
signs. It is counting vectormultiplets minus hypermultiplets, and can lead to
cancellations, and hence it can underestimate the entropy. In the case examined in
[32] it gives the “right” answer, i.e. the answer that coincides with
supergravity.

The Rademacher formula gives an infinite series of subleading corrections

organized by terms in the Farey sequences. In section 2.6 we will discuss the physical interpretation of these subleading corrections.

2.5.3. Poincaré Series for the Elliptic Genus

Finally, let us return to the main task of this section: Expressing the elliptic genus as a Poincaré series in a form suitable to interpretation within the AdS/CFT correspondence.

The manipulations of section 2.5 above have analogs for Jacobi forms. Let denote the space of weak Jacobi forms of weight and index . The analog of the polar part (2.42) is the sum over Fourier coefficients with

Applied to the elliptic genus the relevant Poincaré series becomes:

where is the sum over relatively prime pairs with , while is a finite sum over with , and was defined in (2.31).

In the next section we are going to sketch how this sum can be interpreted as a sum over solutions to 10D supergravity.

Note added, Dec. 8, 2007: Don Zagier pointed out an important error in versions 1-3 of this paper. The map

with

does not map Jacobi forms , contrary to what was asserted in versions 1-3. Nevertheless, for , the can be obtained as Fourier coefficients from the Poincaré series (2.62). For further details see the corrected version 3 of [2], as well as [33], which writes a regularized Poincaré series for the elliptic genus itself, and not its “Fareytail transform.”

2.6. AdS/CFT Interpretation of the Poincaré Series

In the previous section we wrote down the Poincaré series (2.62) for the elliptic genus. This is a mathematical fact, and we are regarding this exact result as a precious piece of “experimental data” to tell us how to formulate the string theory side of the AdS/CFT correspondence. As we will see, the precise formulation of string theory on is full of interesting subtleties. We will now proceed to interpret the various factors in (2.62) in physical terms.

2.6.1.Average over and BTZ black holes

We are going to describe the AdS dual to a conformal field theory computation of a partition function. Therefore, the conformal boundary of the should be a torus. Therefore, we will be looking at 3-dimensional geometries filling in . The metric will accordingly have boundary conditions:

for . Here , , and determines the conformal structure of the torus at infinity.

The only smooth complete hyperbolic geometry satisfying these conditions has the topology of a solid torus. One way to realize this geometry is to take a quotient of the upper half plane by the group ZZ acting as . We can compactify the space by adding the boundary at infinity . We must omit to get a properly discontinuous group action. Topologically, the resulting space is a solid torus.

While the hyperbolic geometry is unique, in order to do physics we need to make a choice of what is called “space” and what is called “time” in the torus at infinity. This choice will affect computations of action, entropy etc. It is this choice which accounts for the sum over , that is, over relatively prime integers in (2.62). Geometrically, describes the unique primitive homology cycle which becomes contractible upon filling in the torus with a solid torus.

For example, let us choose coordinates on . If we choose the term then it is the “spatial” -circle which is filled in. In this case the geometry has the interpretation of an “AdS gas” – that is, we analytically continue the time in Lorentzian AdS and identify it with .

On the other hand, in the term corresponding to it is Euclidean “time” - the -circle - which is filled in. In this case we have the Euclidean “BTZ black hole.” Note that the spatial circle is noncontractible: There is a hole in space, and it is in fact correctly interpreted as a true black hole solution of gravity, as shown in great detail in [34][35].

The general solution is labelled by a point in

and is labelled by the homology class of the primitive cycle
which is contractible.
This family of black holes is the proper interpretation of
what Maldacena and Strominger
termed an “ family of black holes” in
[36]. Thus, the first, and most basic aspect of
(2.62) is that it is a sum over this family of black holes
(including the AdS gas ).
^{†}^{†} An heuristic version of this sum was first written down in
[36].

2.6.2.Low energy Chern-Simons theory

Now, we would like to compute the contribution of the string theory path integral to each term in the sum over pairs in (2.62). A crucial point is that the elliptic genus is unchanged under deformation of parameters. This allows us to focus on the low energy and long-distance limit of the reduction of 10d supergravity on . In this limit, the dominant term in the supergravity action is that of a Chern-Simons theory. The Chern-Simons supergroup is [37]

and the explicit action is

The connections are derived from the negative curvature metric via where is the spin connection and is the dreibrein [38][35]. The gauge fields arise from Kaluza-Klein reduction on . For a detailed derivation of these terms in the action see [39][40][41].

We must choose boundary conditions for the Chern-Simons gauge fields. The boundary values of the connections for , and couple to CFT left- and right-movers, respectively. The boundary conditions (2.65) determine boundary conditions on the gauge fields. In addition: The gauge fields become flat at infinity and the proper boundary conditions are:

where

Because of our choice of fermion spin structures the boundary conditions of the right-moving gauge fields should drop out. This point deserves to be understood more fully.

2.6.3. Spinning in 6-dimensions

Actually, we have not yet fully enumerated the distinct types of geometry that we must sum over. When we include the -dependence in the elliptic genus it is necessary to consider six-dimensional geometries. This leads to an interpretation of the sum on in (2.62).

The BTZ black holes have natural generalizations to quotients of the form

with ZZ acting on by

These correspond to solutions spinning in six dimensions with . Such solutions have been nicely described in detail in [42]. Closely related smooth solutions associated with BPS states have been described in [43].

In the effective Chern-Simons theory these solutions correspond to the insertion of a Wilson line in the center of the solid torus as in Fig. 2. Since the theory is governed by a Chern-Simons theory we expect to see the wavefunction associated to such theories in the partition function. It is well-known that these wavefunctions are given by the affine Lie algebra characters of level current algebra for spin . Another basis of wavefunctions count states at definite values of . These are given by level theta functions:

Fig. 2: A black hole spinning in 6 dimensions is effectively equivalent to the partition function on a solid torus with a Wilson line insertion.

To summarize, we can interpret the contribution of and as a BTZ black hole with homology class contractible and with Wilson lines inserted so that the Chern-Simons wavefunction has definite values of modulo , as in fig. 2.

2.6.4.The light particles of supergravity

Let us now interpret the sum over the polar part in (2.62),

In order to do this we must address some aspects of the Lorentzian version of the AdS/CFT correspondence.

In the Lorentzian version, there is an isomorphism of Hilbert spaces between the Hilbert space of the boundary conformal field theory and some much more mysterious Hilbert space of quantum gravity (string theory) on some interior space. The Hilbert space of the conformal field theory is rather well-understood. We will view it as a Hilbert space graded by the values of . In the elliptic genus, the left-moving Ramond sector states have quantum numbers which we identify as the eigenvalues

Now, we expect such states to correspond to states in the quantum gravity Hilbert space. Symmetry principles (i.e. matching of superconformal symmetries) show us that we must interpret as the 2+1 dimensional energy + spin, while should be viewed as the eigenvalue for spin in the directions.

From the point of view of quantum gravity, there is an important distinction between states which are small perturbations on an AdS background - we will refer to these as “particle states” - and states which form black holes. The distinction is governed by the “cosmic censorship bound” [44][45][42]. Black holes correspond to semiclassical states in . The corresponding states in have in the Ramond sector related to the mass of the black hole by [46]. On the other hand, the condition for a black hole to have a nonsingular horizon is [44][45][42]. Such states therefore have . Thus the unitarity region in the plane is divided into two regions: Supergravity states with are not sufficiently massive to form black holes, corresponding to the shaded region in Fig. 3, while states with will form black holes. Thus, the states which do not form black holes correspond precisely to the to the polar part of the Jacobi form! Moreover, the degeneracy is precisely that of right-BPS supergravity particles from Kaluza-Klein reduction of supergravity on [37].

Fig. 3: The states in the shaded region are not sufficiently energetic to form black holes. These states have quantum numbers corresponding to the polar part of the elliptic genus. Note that quantum numbers not on the axis are not BPS states. The discussion above pertains to states which are right-BPS.

2.6.5. Gravitational action and final factor

According to our interpretation, the final factors

should arise from a careful evaluation of an analytic continuation of Chern-Simons theory to Euclidean signature.

Thus one is naturally let to attempt a careful evaluation of the gravitational action for the spinning extremal black holes. The Einstein action is

where is the second fundamental form of the boundary. Since the Einstein action on is infinite it must be regularized. The standard way to do this is to introduce a boundary, thus necessitating the second term. The difference of such actions between two geometries in the family (2.66) can be evaluated in a well-defined way and gives:

Moreover, the computations of [42] produce such an entropy factor weighted by in the six-dimensional case.

Upon taking a limit the expression (2.76) closely resembles (2.74), but, so far as we know, there is no honest and convincing derivation of (2.74) in the literature starting from the Chern-Simons approach.

Note that (2.74) is odd under . This is the reason we must put a restriction on the Poincaré series (2.62). The transformation corresponds to the diffeomorphism on the boundary torus. The fact that the summand in (2.62) should be understood better. Perhaps it is due to the fact that only Ramond groundstates contribute.

2.7. Summary: Lessons & Enigmas

We have presented some evidence to suggest that the full AdS-interpretation of the elliptic genus of the boundary conformal field theory can be expressed in the form

where is a wavefunction for a Chern-Simons theory and where the sum is over Euclidean solutions of supergravity of spinning black holes with supergravity particles in . It should be clear to the reader that there are gaps and enigmas in this story. For examples,

1. Why do we need to take the Serre dual to get a reasonable formula?

2. What is the origin of the factor

from the string partition function? Note that this factor is crucial for the convergence of the sum over . It also has the pleasant property that is a well-defined half-density on the universal elliptic curve.

3. Is it sufficient to focus purely on the Chern-Simons sector to evaluate the path integral or must one take into account the full tower of string fields? (We have been assuming the latter contribute a trivial factor to , because of its topological nature.)

4. Perhaps the most important enigma is the origin of the sum over the polar part in (2.62). This is probably saying something significant about the Hilbert space of quantum gravity. It indicates that the nature of the isomorphism between the CFT Hilbert space and the string theory Hilbert space is qualitatively different for the infinite set of conformal field theory states above the cosmic censorship bound. What replaces a sum over states in the Euclidean quantum gravity Hilbert space is a sum over a special set of geometries. Note in particular that the term does not contribute. These are the unique quantum numbers (the so-called “ BTZ” black hole) of states which are simultaneously topological and black holes. It is possible that this structure is related to the phenomenon of “asymptotic darkness” that has been advocated by T. Banks [47][48].

2.8. Applications

Whether or not one believes the physical interpretation advocated in the previous section, the formula (2.62) is true, and has some some nice applications.

One application is to the thermodynamics of string theory on Euclidean . One discovers a 3-dimensional version of the deconfining phase transition of large Yang-Mills theory discussed by Witten [49]. In the case one studies the partition function as a function of

where is the spin fugacity and is the inverse temperature. In the large limit becomes a piecewise analytic function of . It is simplest to study the partition function in the sector (by setting ). As at fixed the dominant geometry is characterized by the pair which maximizes

This geometry contributes a term of order

The standard keyhole region fundamental domain for has the property that the modular image of any point has an imaginary part . Therefore, the phase domains are given by and its modular images.

As a second application we note that a computation similar in spirit to what we have discussed was performed by Maldacena to resolve a sharp version of the “black hole information paradox” for eternal AdS black holes. See [50].

2.9. Speculations on future applications of AdS/CFT to number theory

In this section we present some speculations on ways in which the AdS/CFT correspondence might have some interesting interactions with number theory. Our speculations are based on ongoing discussions with A. Strominger, and have at times involved B. Mazur, and S. Gukov. For some related ideas see [3]. (Some overlapping remarks were made recently in [51][52].)

2.9.1. Quotients of AdS/CFT

Suppose string theory on is dual to a conformal field theory . Suppose

is an infinite discrete group. Since acts as a group of isometries in the bulk theory, we can consider string theory on

It is natural to ask if string theory on (2.83) makes sense, and if so, whether it is dual to some kind of “quotient” of the conformal field theory by . Note that such a quotient, if it even exists, is very different from an orbifold of a conformal field theory, for acts by conformal transformations on the “worldsheet” rather than the “target space” of .

Such a duality, if it were to make sense, would have very interesting implications in at least two ways. First, there would be important applications to questions of cosmology and time dependence in string theory. Second – and more central to the theme of these lectures – there would be interesting applications to number theory. In the following sections we will sketch some of the possible applications.

The reader should be warned at the outset that there are nontrivial difficulties with the idea that AdS/CFT duality can survive general quotients by such groups . The difficulties stem from the fact that the “interesting” groups we wish to consider act on the conformal boundary at infinity, , but the action is sometimes ergodic. More precisely, the boundary is divided into a disjoint union of two regions:

The first region is the domain of discontinuity. Here the group acts propertly discontinuously and the quotient is, for , a Riemann surface. Note that this Riemann surface can have cusps and several connected components. The complementary region is called the limit set. It is the closure of the set of accumulation points of , and the action on is ergodic. This means that any “quotient” of the boundary conformal field theory is going to have strange behavior on . To take an extreme example, there are groups with no domain of discontinuity. Then the classical quotient is a compact hyperbolic manifold. So the “boundary theory,” if it exists, must surely be something truly unusual.

In fact, the quotient by can produce strange causal structure in the Lorentzian case, a fact which probably indicates large backreaction in the context of supergravity. A related point is that the distance between image points can get small, again indicating breakdown of the sugra approximation. Indeed, the existence of a boundary theory for groups with nontrivial limit set has been argued against by Martinec and McElgin [53][54].

Nevertheless, a successful outcome would undoubtedly lead to many very fascinating things, so let us suppose that a dual boundary theory does exist and briefly ask what it might be good for.

2.9.2. String Cosmology

A few years ago, in [55], interesting cosmologies with singularities were considered based on spacetimes of the form (2.83).

More recently, string theory with time-dependent singularities in “soluble” string models has come under some scrutiny. Amongst the many investigations in this area is the work in [56][57][58][59] which studies the ZZ-orbifold of defined by the action

where are light-cone coordinates. It turns out that string perturbation theory in such backgrounds is highly problematic. The difficulties are expected to be a generic feature of strings in cosmological singularities. Moreover, nonperturbative effects involving black holes are expected to be important [60]. This is relevant to the present discussion for the following reason. Recall that is the universal covering space . The Lie algebra is Minkowski space. Consider the action on by ZZ with

where is a parabolic element. In the scaling region of these look like the cosmological models (2.85). On the other hand, since there is a boundary theory summarizing all the nonperturbative physics, it is reasonable to think, provided the AdS/CFT correspondence survives the quotient construction, that the boundary theory contains some clue as to the resolution of the cosmological singularity. Some investigations along these lines were carried out in [61], but there is much more to understand.

2.9.3. Potential Applications to Number Theory: Euclidean version

One of the possible applications of these ideas to number theory concerns the theory of modular symbols.

Let us recall (in caricature) the computation of the 2point function of spinless primary fields. In AdS the tree-level 2-point function of scalar fields is the Green’s function:

In we have the simple explicit formula:

where

One extracts the 2point correlator from the boundary behavior of the Green’s function:

as . This leads to the familiar result:

where is the dual operator of (2.12).

Now, let be discrete and suppose AdS/CFT “commutes with orbifolding.” In the tree-level approximation, the Green’s function on is obtained by the method of images. Therefore, according to (2.90) the boundary CFT correlator should be obtained from the method of images. For a primary field (with spin ) of weights this would lead to

We would like to stress that in general in CFT it is not true that
the conformal correlators on Riemann surfaces are
obtained by the method of images. While it is true that the Green’s function of
a scalar field is obtained by summing over images, in the presence of
interactions there are further correlations between a source and its image point.
^{†}^{†} As a simple example, if is a free massless scalar field then
is a sum of images, and therefore
is a product over images!
Therefore, at best (2.92) can apply in the large
approximation (which justifies the tree-level supergravity). Even there,
AdS/CFT is making a highly nontrivial prediction for the boundary CFT
correlators.

Nevertheless, let us accept (2.92). Now suppose there is a flat gauge field in the low energy supergravity coupling to charged scalars . Then the boundary correlator becomes: