Homework Problems 609

PROBLEM 11.13

Verify the result for the partition function (11.162).

PROBLEM 11.14

Derive Eq. (11.194).

PROBLEM 11.15

Wald™s formula determines the entropy of a D = 4 black hole when the e¬ec-

tive action contains terms of higher order in the curvature tensor. Denoting

the e¬ective Lagrangian density by L, Wald™s formula expresses the entropy

as an integral over the horizon of the black hole

‚L

d2 „¦.

S = 2π µµν µρ»

‚Rµνρ»

S2

Verify that Wald™s formula gives the usual BH entropy formula when only

the Einstein“Hilbert term is present.

PROBLEM 11.16

Perform microstate counting to obtain the entropy of the nonextremal three-

charge black hole given in Eq. (11.69).

PROBLEM 11.17

Perform microstate counting to obtain the entropy of the D = 5 rotating

black hole given in Eq. (11.76). Also, derive the entropy formula given in

Eq. (11.84).

12

Gauge theory/string theory dualities

Many remarkable dualities relating string theories and M-theory have been

described in previous chapters. However, this is far from the whole story.

There is an entirely new class of dualities that relates conventional (non-

gravitational) quantum ¬eld theories to string theories and M-theory.

There are three main areas in which such a gauge theory/string theory

duality emerged around the mid to late 1990 s that are described in this

chapter:

• Matrix theory

• Anti-de Sitter/conformal ¬eld theory (AdS/CFT) correspondence

• Geometric transitions

Historically, string theory was introduced in the 1960 s to describe hadrons

(particles made of quarks and gluons that experience strong interactions).

Strings would bind quarks and anti-quarks together to build a meson, as

depicted in Fig. 12.1 or three quarks to make a baryon. As this approach was

developed, it gradually became clear that critical string theory requires the

presence of a spin 2 particle in the string™s spectrum. This ruled out critical

string theory as a theory of hadrons, but it led to string theory becoming

a candidate for a quantum theory of gravity. Also, QCD emerged as the

theory of the strong interaction. The idea that there should be some other

string theory that gives a dual description of QCD was still widely held,

but it was unclear how to construct it. Given this history, the discovery

of the dualities described in this chapter is quite surprising. String theory

and M-theory were believed to be fundamentally di¬erent from theories

based on local ¬elds, but here are precise equivalences between them, at

least for certain background geometries. In fact, it seems possible that

every nonabelian gauge theory has a dual description as a quantum gravity

theory. To the extent that this is true, it answers the question whether

610

Gauge theory/string theory dualities 611

quantum mechanics breaks down when gravity is taken into account with

a resounding no, because the dual ¬eld theories are quantum theories with

unitary evolution.

q q

Fig. 12.1. A meson can be viewed as a quark and an antiquark held together by a

string.

The methods introduced in this chapter can be used to study the infrared

limits of various quantum ¬eld theories. Realistic models of QCD, for exam-

ple, should be able to explain con¬nement and chiral-symmetry breaking.

These properties are not present in models such as N = 4 super Yang“Mills

theory due to the large amount of unbroken supersymmetry. There is a vari-

ety of ways to break these symmetries so as to get richer models, in both the

AdS/CFT and geometric transition approaches. In this setting, phenomena

such as con¬nement and chiral-symmetry breaking can be understood.

Matrix theory

With the discovery of the string dualities described in Chapter 8, it became

a challenge to understand M-theory beyond the leading D = 11 supergrav-

ity approximation. Unlike ten-dimensional superstring theories, there is no

massless dilaton, and therefore there is no dimensionless coupling constant

on which to base a perturbation expansion. In short, 11-dimensional su-

pergravity is not renormalizable. Of course, ten-dimensional supergravity

theories are also not renormalizable, but superstring theory allows us to

do better. So one of the most fundamental goals of modern string the-

ory research is to understand better what M-theory is. An early success

was a quantum description of M-theory in a ¬‚at 11-dimensional space-time

background, called Matrix theory. This theory is discussed in Section 12.2.

Its fundamental degrees of freedom are D0-branes instead of strings. The

generalization to toroidal space-time backgrounds is also described. Matrix

theory is formulated in a noncovariant way, and it is di¬cult to use for ex-

plicit computations, so the quest for a simpler formulation of Matrix theory

or a variant of it is an important goal of current string theory research.

Nevertheless, the theory is correct, and it has passed some rather nontrivial

tests that are described in Section 12.2.

612 Gauge theory/string theory dualities

AdS/CFT duality

By considering collections of coincident M-branes or D-branes, one ¬nds a

space-time geometry that has the features discussed in Chapter 10. The

branes are sources of ¬‚ux and curvature, and a warped geometry results. In

certain limits the gauge theory on the world-volume of the branes describes

precisely the same physics as string theory or M-theory in the warped ge-

ometry created by the branes. In this way one is led to a host of remarkable

gauge theory/string theory dualities.

In their most straightforward realization, AdS/CFT dualities relate type

IIB superstring theory or M-theory in space-time geometries that are asymp-

totically anti-de Sitter (AdS) times a compact space to conformally invariant

¬eld theories.1 Anti-de Sitter space is a maximally symmetric space-time

with a negative cosmological constant. Even though it is spatially in¬nite in

extent, one can de¬ne a boundary at in¬nity. For reasons to be explained,

the space-time manifold of the conformal ¬eld theory (CFT) is associated

with this boundary of the AdS space. Therefore, these are holographic du-

alities. The name is meant to re¬‚ect the similarity to ordinary holography,

which records three-dimensional images on two-dimensional emulsions.

The conjectured AdS/CFT correspondences are dualities in the usual

sense: when one description is weakly coupled, the dual description is

strongly coupled. Thus, assuming that the conjecture is correct, it allows the

use of weak-coupling perturbative methods in one theory to learn nontrivial

facts about the strongly coupled dual theory. Just as Matrix theory can be

regarded as de¬ning quantum M-theory in certain space-time backgrounds,

a possible point of view is that the AdS/CFT dualities serve to complete

the quantum de¬nitions of string theories and M-theory for another class

of space-time backgrounds. Ideally, one would like to have a background-

independent de¬nition of these quantum theories, but that does not exist

yet. Even so, what has been achieved is really quite remarkable.

The AdS/CFT conjecture emerged from considering the space-time ge-

ometry in the vicinity of a large number (N ) of coincident p-branes. The

three basic examples of AdS/CFT duality, which have maximal supersym-

metry (32 supercharges), correspond to taking the p-branes to be either

M2-branes, D3-branes, or M5-branes. The corresponding world-volume the-

ories (in three, four, or six dimensions) have superconformal symmetry, and

therefore they are superconformal ¬eld theories (SCFT). In each case the

dual M-theory or string-theory geometry is the product of an anti-de Sitter

space-time and a sphere:

1 The conformal group in D dimensions was de¬ned in Chapter 3.