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Introduction to AdS-CFT

lectures by

Horatiu Nastase

Global Edge Institute, Tokyo Institute of Technology

Abstract

These lectures present an introduction to AdS-CFT, and is in-tended both for begining and more advanced graduate students,which are familiar with quantum field theory and have a workingknowledge of their basic methods. Familiarity with supersymmetry,general relativity and string theory is helpful, but not necessary, asthe course intends to be as self-contained as possible. I will intro-duce the needed elements of field and gauge theory, general relativ-ity, supersymmetry, supergravity, strings and conformal field the-ory. Then I describe the basic AdS-CFT scenario, of N = 4 SuperYang-Millss relation to string theory in AdS5S5, and applicationsthat can be derived from it: 3-point functions, quark-antiquark po-tential, finite temperature and scattering processes, the pp wavecorrespondence and spin chains.

1

1 Elements of quantum field theory and gauge theory

Here I will review some elements of quantum field theory and gauge theory that will beneeded in the following.

The Fenyman path integral and Feynman diagramsConventions: throughout this course, I will use theorists conventions, where = c = 1.

To reintroduce and c one can use dimensional analysis. In this conventions, there is only onedimensionful unit, mass = 1/length = energy = 1/time = ... and when I speak of dimensionof a quantity I refer to mass dimension, i.e. the mass dimension of d4x, [d4x], is 4. TheMinkowski metric will have signature ( + ++), thus = diag(1,+1,+1,+1).

I will use the example of the scalar field (x), that transforms as (x) = (x) under acoordinate transformation x x. The action of such a field is of the type

S =

d4xL(, ) (1.1)

where L is the Lagrangian density.Classically, one varies this action with respect to (x) to give the classical equations of

motion for (x)L

= L

()(1.2)

Quantum mechanically, the field (x) is not observable anymore, and instead one mustuse the vacuum expectation value (VEV) of the scalar field quantum operator instead, whichis given as a path integral

< 0|(x1)|0 >=

DeiS[](x1) (1.3)

Here the symbol D represents a discretization of spacetime followed by integration ofthe field at each discrete point:

D(x) =i

d(xi) (1.4)

A generalization of this object is the correlation function or n-point function

Gn(x1, ..., xn) =< 0|T{(x1)...(xn)}|0 > (1.5)The generating function of the correlation functions is called the partition function,

Z[J ] =

DeiS[]+i

d4xJ(x)(x) (1.6)

It turns out to be convenient to write quantum field theory in Euclidean signature, andgo between the Minkowski signature (+ ++) and the Euclidean signature (+ + ++) via aWick rotation, t = itE and iS SE , where tE is Euclidean time (with positive metric)and SE is the Euclidean action.

2

The partition function in Euclidean space is

ZE[J ] =

DeSE []+

d4xJ(x)(x) (1.7)

and the correlation functions

Gn(x1, ..., xn) =

DeSE [](x1)...(xn) (1.8)

are given by differentiation of the partition function

Gn(x1, ..., xn) =

J(x1)...

J(xn)

DeSE []+

d4xJ(x)(x)|J=0 (1.9)

This formula can be calculated in perturbation theory, using the so called Feynmandiagrams. To exemplify it, we will use a scalar field Euclidean action

SE[] =

d4x[

1

2()

2 +m22 + V ()] (1.10)

Here I have used the notation

()2 =

= = 2 + ()2 (1.11)

Moreover, for concreteness, I will use V = 4.Then, the Feynman diagram in x space is obtained as follows. One draws a diagram,

in the example in Fig.1a) it is the so-called setting Sun diagram.A line between point x and point y represent the propagator

(x, y) = [ +m2]1 =

d4p

(2)4eip(xy)

p2 +m2=

1

(x y)2 (1.12)

A 4-vertex at point x represents the vertexd4x() (1.13)

And then the value of the Feynman diagram, F(N)D (x1, ...xn) is obtained by multiplying

all the above elements, and the value of the n-point function is obtained by summing overdiagrams, and over the number of 4-vertices N with a weight factor:

Gn(x1, ..., xn) =N0

1

N !

diag D

F(N)D (x1, ..., xn) (1.14)

(Equivalently, one can use a 4/4! potential and construct only topologically inequivalentdiagrams and the vertices are still

d4x(), but we now multiply each inequivalent diagram

by a statistical weight factor).

3

k1p2p3

x1 x2 x3 x4

a)

k1 p1

p2

p3

b)

d=2 d=4

d)

d=6

e)c)

Figure 1: a) Setting sun diagram in x-space. b) Setting sun diagram in momentum space.c)anomalous diagram in 2 dimensions; d)anomalous diagram (triangle) in 4 dimensions;e)anomalous diagram (box) in 6 dimensions.

We mentioned that the VEV of the scalar field operator is an obsevable. In fact, thenormalized VEV in the presence of a source J(x),

(x; J) =J < 0|(x)|0 >J

J < 0|0 >J =1

Z[J ]

DeS[]+J (x) =

JlnZ[J ] (1.15)

is called the classical field and satisfies an analog (quantum version) of the classical fieldequation.

S matricesFor real scattering, one constructs incoming and outgoing wavefunctions, representing

actual states, in terms of the idealized states of fixed (external) momenta k.Then one treats the scattering of these idealized states and at the end one convolutes

with the wavefunctions. The S matrix defines the transition amplitude between the idealizedstates by

< p1, p2, ...|S|k1, k2, ... > (1.16)The value of this S matrix transition amplitude is given in terms of Feynman diagramsin momentum space. The diagrams are of a restricted type: connected (doesnt containdisconnected pieces) and amputated (which means that one does not use propagators for theexternal lines).

For instance, the setting sun diagram with external momenta k1 and p1 and internalmomenta p2, p3 and k1 p2 p3 in Fig.1b) is

4(k1 p1)d4p2d

4p32 1

p22 +m2

1

p23 +m2

1

(k1 p2 p3)2 +m24(1.17)

4

The LSZ formulation relates S matrices in Minkowski space with correlation functionsas follows. The Fourier transformed n +m-point function near the physical poles P 2I = M

2

behaves as

Gn+m(p1, ..., pn)(x1, ..., xn) (

ni=1

Zi

p2i m2 + i

)(mj+1

Zi

k2j m2 + i

)< p1, ..., pn|S|k1, .., km >

(1.18)For this reason, the study of correlation functions, which is easier, is preferred, since any

physical process can be extracted from them as above.If the external states are not states of a single field, but of a composite field O(x), e.g.

O(x) = ()(x)(+...) (1.19)it is useful to define Euclidean space correlation functions for these operators

< O(x1)...O(xn) >Eucl=

DeSEO(x1)...O(xn)

=n

J(x1)...J(xn)

DeSE+

d4xO(x)J(x)|J=0 (1.20)

which can be obtained from the generating functional

ZO[J ] =

DeSE+d4xO(x)J(x) (1.21)

Yang-Mills theory and gauge groupsElectromagnetismIn electromagnetism we have a gauge field

A(x) = ((x, t), A(x, t)) (1.22)

with the field strength (containg the E and B fields)

F = A A = 2[A] (1.23)

The observables like E and B are defined in terms of F and as such the theory has agauge symmetry under a U(1) group, that leaves F invariant

A = ; F = 2[] = 0 (1.24)

The Minkowski space action is

SMink = 14

d4xF 2 (1.25)

which becomes in Euclidean space

SE =1

4

d4x(F)

2 =1

4

d4xFF

(1.26)

5

The coupling of electromagnetism to a scalar field and a fermion field is obtained asfollows

StotalE = SE,A +

d4x[(D/+m) + (D)

D]

D/ = D; D = ieA (1.27)

This is known as the minimal coupling. Then there is a U(1) local symmetry that extendsthe above gauge symmetry, namely

= eie(x); = eie(x) (1.28)

under which D transforms as eieD, i.e transforms covariantly, as does D.

The reverse is also possible, namely we can start with the action for and only, with instead of D. It will have the symmetry in (1.28), except with a global parameter only.If we want to promote the global symmetry to a local one, we need to introduce a gaugefield and minimal coupling as above.

In the following, I will sometimes replace e by ie, thus D = + eA.Yang-Mills fieldsYang-Mills fields Aa are self-interacting gauge fields, where a is an index belonging to a

nonabelian gauge group. There is thus a 3-point self-interaction of the gauge fields Aa, Ab , A

c,

that is defined by the constants fabc.The gauge group G has generators (T a)ij in the representation R. T

a satisfy the Liealgebra of the group,

[Ta, Tb] = fabcTc (1.29)

The group G is usually SU(N), SO(N). The adjoint representation is defined by (T a)bc =fabc. Then the gauge fields live in the adjoint representation and the field strength is

F a = Aa Aa + gfabcAbAc (1.30)

One can define A = AaTa and F = FaTa in terms of which we have

F = A A + g[A, A ] (1.31)(If one further defines the forms F = 1/2Fdx

dx and A = Adx where wedge denotes antisymmetrization, one has F = dA+ gA A).

The generators T a are taken to be antihermitian, their normalization being defined bytheir trace in the fundamental representation,

trT aT b = 12ab (1.32)

and here group indices are raised and lowered with ab.The local symmetry under the group G or gauge symmetry has now the infinitesimal

formAa = (D)

a (1.33)

6

where(D)

a = a + gfabcA