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AdS/CFT

Black Holes

&Matrix Models

Oviedo 9/9/2005

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Over thirty years ago...

Large-N limit ofYM Theories

Hawking processBlack Hole evaporation

It is in the last decade that we began tounderstand that they were talking

about the same thing

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In the early eighties we learned that the fundamental theory of thestrong interactions (QCD) is and AF gauge theory with group

AF implies many things

Dimensional transmutation

Confinement, XSB, no expansion parameters

SU(3)

E g(E) 0

E 0 g(E)

g QCD

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'tHooft

Confined theories may be described byeffective strings (BCS)

Qualitative properties of hadron spectrum

Veneziano-Witten relations, U(1)-problem...

The colour group is SU(3). Imagine there are N-

colors: SU(N). Expand in a dimensionless parameter

1

N

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How does it work ?

Ignore Lorentz indices and space-time dependence

The fields are matrix valued(A)

ij i, j = 1, . . . N

Z(g1, g2,...) =

DMe

N

igiM

i

The propagator is

MijMkl =

Nil

kj

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P= Number of edges

I= Number of faces

V= Number of vertices

NVP+I

= N()

() = 2

2hb (Euler number)

N()

||

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Z= S2 + T2 + T2#T2 + . . .N2 N0 N2

A TOPOLOGICAL EXPANSION

A STRING EXPANSION

gst

1

N g2

YMN=

'tHooft coupling

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MATRIX RELOADED

QCD perturbation theory has a topological expansion in 1/N. Theleading contribution comes from the sphere, then the torus, and

then higher genus Riemann surfaces...

EXACTLY LIKE A STRINGTHEORY EXPANSION

Which String Theory is behind QCD?

=e2

4=

1

137; e

1

3

N=3 A good expansion parameter? Large N provides a compelling

qualitative explanation of hadron properties

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Hawking

The seventies saw the renaissance of General Relativity, BlackHole Theoy, BH thermodynamics, the membrane paradigm...

Bekensetein, Hawking

It is time to do QFT in the presence of a BH

or a collapsing star; as with the semiclassicaltheory of radiation

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Carter Penrose diagrams of star collapse.

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Black Holes are not so black after all

There is black body emission associated to a black hole once weconsider the quantization of fields in its vicinity

TH =hc

3

8GMk

One of the most beautiful and profound formulas in Physics inthe last fifty years

It has an associated Entropy

S=Ah

4AP

Since S goes like the square of M, we find a limiting

temperature

AP =hGN

c3

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No hair theorems L,M,Q

But huge entropy...

How do we count the number of states?

S= logW

Strings Branes Supergravity

Polchinski

Strominger-Vafa

Near horizon Physics can be described with

branes, CFT's with Dirichlet boudary conditions

Branes, plus Sugra states, plus Cardy'sformula provide a real count of statescontributing to the entropy of the hole

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There is a large class of BHsolutions in String/Sugra theories

None of the Strominger-Vafa counts comeclose yet to the Schwarschild entropy count.It is too far from the BPS (partial SUSY)

character

Beyond D=4 it seems that the no-hairtheorem does not apply (black rings, D=5,

Emparan and Reall )

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Enter Maldacena

Large N limits of SCFT and Sugra hep-th/9711200

Some SCFT contain in their Hilbert space a sector describingSUGRA (IIB) on AdS x W

He considered N coincident branes inM/String Theory, in the near horizonlimit and at low energies, where the

brane QFT decouples from the bulk. Inlarge N we can trust the near horizon

geometry

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N->Infinity, N=4 SCFT in D=4 contains in itsHilbert space IIB Strings

IIB on AdS5xS5N units of RR fluxis equivalent to

N=4 SCFT on the boundaryS4 of M4

Holography at its best but with a vengeance

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gst gY M 0

N

R (g2YMN)1/4 Large

The sugra description is an

accurate characterization ofString Theory

Correlators

e

S4

0O = ZS(0)

Conformal dimensions are the masses of the sugra multiplets

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Careful scrutiny of chiral superfields in N=4

SYM in D=4 matches masses of KK harmonicsin AdS5xS5

BMN limit, involving angular momentum, PP

wave backgrounds

Lower dimensional examples AdS3xS7...related to well studied CFT's in D=2

Many tests of the conjecture, but no theorem...

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General conjecture & consequences

AdS/CFT relates CFT on Sd

to AdS(d+1)xW, with W a compact manifold

Replace Sd

M

Then replace AdS(d+1) by an Einstein manifold X with < 0

Instead of d+1 dimensional manifold X, use a manifold Y

with dimension 10,11, looking near infinity as X x W forsome Einstein space X.

ZCFT(M) = exp(IS(X))

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If there are several X's, sum

ZCFT =i

eN

F(Xi)

In the large N limit IS(Xi) N, > 0; = 2 for N=4 SUSY

We can encounter large-N phase transitions

F(M) = logZCFT(M)

limN

F(M)

N= F(Xi)

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Confinement-Deconfinement phasetransition

Matrix revolutions

M = S1 S

d1

Finite temperature QFTConfinement-Deconfinement Transition

Hagedorn transition

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To make a long story short, we can construct two partitionfunctions:

Z1(M) = TreH

(

1)F

Z2(M) = TreH

There are two manifolds X1, X2 satisfying the above conditions

AdS(X1)ds

2 = (1 + r2)dt2 +dr

2

1 + r2+ r2d2

AdS BH(X2)ds

2 = V(r)dt2 +dr

2

V(r)+ r2d2

V(r) = 12m

r

+r2

b2

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V(r+) = 0 Horizon condition

=

12r+1 + 3r2+ =

4

V(r+)

X1 contributes to Z1, Z2X2 Contributres to Z2

Depending on the parameters and the temeperature we can pass from X1 toX2 dominance. The associated phase transition is presicely related to the

confinement-deconfinement transition

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A meaningful plot:

In AdS there is a minimun temperature, below, there is no BH.Above we find two black holes SBH and BBH

SBH is unstable, a bounce solutionBBH is stable with positve heat capacity. Line I corresponds tothe formation of the black holes, line II to the Hawking-Pagetransition, where the BBH is more stable than hot AdS, and line

III corresponds to the Horowitz-Polchinski transition.

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Can we reproduce HP with pure gauge computations?

YES

S3 S

1QCDR

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Matrix Galore

S(U,U) =

i=1

aitrUitrUi+

+

kk

kkk(U)k(U

1)

k(U) =

j

(trUj)kj k,k =

k,k

The large N analysis of this action does reproduce the HPtransitions and the Hagedorn transition (also the Horowitz-

Polchinski transition)

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Confinement-Deconfinement are clearly related togeometric transitions (matching of parameters,relevant and marginal)

Full and consistent computation of BH nucleationrates (H.Liu, C.. Gomez, S. Wadia, L.A-G. P. Basu, M.Marino...) no IR problems.

Computation of critical exponents at all transitions

Unfortunately, large-N phase transitions. The onlypossible ones at finite N

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SUSY is important, but perhaps not so much, we areat finite T

Is there a similar, reliable, holographic description

without SUSY?

At the end we want a large volume theory

The SBH evaporation can be described

holographically in terms of a unitary QFT. Thereshould be no information loss.

MATRIX DEBUGGED

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Please remove time independent states (i.e.equilibrium states)

What is the holographic shadow of a time dependent

collapse?

QFT/String Theory are very poorly understood intime dependent conditions

Learn from RIHC, ALICE, Lattice simulations?

There is hope, Hawking paid its bet!!

MATRIX REPROGRAMMED