RECENT DEVELOPMENTS IN THE GAUGE/GRAVITY CORRESPONDENCE

ALBERTO ZAFFARONI

PadovaDicembre 2007

OUTLINE

1. The AdS/CFT correspondence: a status report

2. Realistic AdS/CFT?

3. Properties of Strongly Coupled Gauge Theories

I. ADS/CFT CORRESPONDENCE: A STATUS REPORT

The AdS/CFT correspondence is the explicit realization of the old idea that a strongly coupled gauge theory at large N has a dual description in terms of an effective string theory.

As suggested, for example, by Feymann diagram analysis at large N.

String theory was born to explain strong interaction: today we know that

• Asymptotically flat backgrounds contains dynamical gravity • Curved and warped backgrounds may be dual to gauge theories

HOLOGRAPHY AND QCD

In principle, it applies to all gauge theory engeneered with branes

Dual gravitational background

Excitations=gravitons etc… representmesons and glueballs of the dualgauge theory.

3 branes backreact on geometry:

N=4 SYM on : black 3 branes in type II B

N=4 SYM in 4d: D3 branes in type II B

3 1S×

55AdS S×

In general, we obtain a metric with a non trivial warp factor

It works with HOLOGRAPHY:

AdS5

CFT

conformal case AdS like throat-like RS model

non conformal caseminimal warp factor

STRING

Gauge theory

Number of dof of quantum gravityIn a region increases with area

(and not volume)

IR Cut-off

u u

0u

GAUGE INVARIANT OPERATOR GRAVITY FIELD

( )O x ( )xω

Extend from boundary using eqs motion

( )xω ( , )x rω

4 5( ) ( ) ( ( , ))CFT AdS

O x x d x L x r d x

e eω ω∫ ∫

< >=GREEN FUNCTIONS FOR O

It is QUANTITATIVE: it computes explicitely mass spectrum of mesons and glueballs and Green functions.

SUPERGRAVITY FIELDS

gµν Tµν

Aµ...Jµ µψγ ψ= +

φ F F µνµν

HOLOGRAPHY:

Just extend a field from boundary

Equation of motion

Bocome Schrodinger equation for spectrum of bound states

Continuum spectrum for conformal theoriesDiscrete spectrum of bound states for confining theories

But computation with a dual effective theory are useful if the dual is weakly coupled. String theory becomes weakly coupled iff

11

s

s

gM R

1/1/( )

s

s YM

g NM R g N∼∼

BUT

This means large N AND strong coupling

Many examples of strongly coupled conformal gauge theories;N=4 SYM has a tunable coupling constant.

QCD is asymptotically free in the UV, so it does not fit in thispicture: its real dual is a strongly coupled string theory.

Many strongly coupled gauge theories becomes EXACTLY SOLVABLE via AdS/CFT

The most studied case is N=4 SYM, where we can combine high supersymmetry,integrability and other features to obtain all orders results also for non protected quantities

Correlation functionsWilson loopsSpectrum of anomalous dimensions

EXAMPLE: the cusp anomalous dimension (dimension high spin operators, planar gluon amplitutes)

weak

strong

In the last decade, many deformations of QCD (with extra fields or modifications at the cut-off) have been constructed which are exactly solvable by AdS/CFT; they are strongly coupled in all regimes.

•Massive Deformations of N=4 SYM

•Compactification of Higher Dimensional SYM

•Cascading Gauge Theories

Massive Deformations of N=4 SYM

Based on the interpretazion of the radial coordinate of AdS as an energy scale

r

Backgrounds that are asympotically AdS for large r correspond to deformazions(or different vacua) of the original theory

UV large r IR small r

C theoremMassive deformations of N=4 SYM

Girardello,Petrini,Porrati,AZFreedman,Gubser,Pilch,Warner

Polchinski,Strassler

Compactification of Higher Dimensional SYM

N=4 SYM on : black 3 branes in type II B3 1S×

Witten black hole.Original example of non susy non conformal theorySpectum of glueball computable

Cascading Gauge Theories

Best example of regular N=1 SYM gauge theory based on Seiberg duality

... ( ) ( ) ( ) ( ) ... ( )SU N M SU N SU N SU N M SU M− > + × − > × − − > − >

Klebanov,Strassler

Strongly coupled at all scaleFancy UV completionReduce to N=1 SYM in the IR (plus Goldstone boson)

( ) ( )SU SU∞ × ∞

Two regular known examples of N=1 gauge theory duals: Klebanov Strassler and Maldacena-Nunez; a continuous family of solutions intepolates between themButti,Grana,Minasian,Petrini, AZ,

A possible message for realistic YM: string tension ratio ( / )( / )

k

p

T Sin k NT Sin p N

ππ

≈

II. REALISTIC AdS/CFT?

Can we study Flavor Physics?

D3 brane engeneering give adjoint or bifundamental fields

Flavors can be added by introducing other typoe of branes (D7,…) which inhibits the near horizon geometry

Flavors can be added using probes (neglecting back reaction)Quenched approximation: including glue loops but not quark loops

Some interesting fully backreacted backgrounds (Nunez and co)

1) Resummation of higher derivatives corrections is in principle doable in string theory: it can be done in flat space and selected backgrounds. Only a technical problem prevent solution of QCD at large N

3) Interesting application to non equilibrium physics:Hydrodynamic properties of hot dense QCD plasma

(shear viscosity, jet quencing parameter -- RHIC).

Strongly coupled: no perturbation theory!Time dependent process: no lattice simulation!

HOWEVER:

2) AdS/QCD: attempt to provide an approximate effective lagrangian forhadron structure with confinement at large distances, almost conformal behaviorat short distances. (Pomarol, Katz, Karch, Brodsky,…)

Nc

Nc g2YM

LATTICE

ADS/CFT

PERTURBATIVEREGION

α’ corrections

Nc

Nc g2YM

ADS/CFT

PERTURBATIVEREGION

Non equilibrium – Time dependent processes

III. Properties of Strongly Coupled Gauge Theories

Proof of a c theorem for strongly coupled gauge theories with AdS dual (c=a)

Simplification of OPE at strong coupling

Shear viscosity bounds

Control on the spectrum of states:

in particular free energy and partition functions for BPS states

The near horizon geometry is

The worldvolume theory is a 4d conformal gauge theory

5AdS H×

CY condition implies that H is Sasaki-Einstein.

Few metrics known ( )

Many interesting question solved without knowledge of the metric

5 1,1 , , ,, , ,p q p q rH S T Y L=

We have a large class of 4d CFT with AdS DUAL: D3 branesprobing Calabi-Yau singularities

EXAMPLES:

N=4 SYM

Conifold1 ,1C (T )

3 5C =C(S )

ij pq i p j qW A B A Bε ε=

A

B

33/C Z Orbifold

ProjectionOf N=4 SYM

152( )C L

U V

W ijk i j kW U V Wε=

...W =

GENERAL RESULTS:

Correspondence between CY and CFT

Spectrum of dimensions and counting of BPS states

•Connections to dimers: (Okunkov,Nekrasov,Vafa – Hanany,Kennaway)

•Geometric computation without metric (Martelli,Sparks,Yau)

•AdS/CFT, combinatorics, a-maximization

Non conformal models, susy breaking,model building

152L

Connection to dimers:Okounkov,Nekrasov,Vafa – Franco,Kennaway,Hanany,Vegh,Wecht

21Y

2 1YdelPezzo 1 =

Dimers, combinatorics and charges:Hanany-Witten construction for local CY

Central charge of the CFT determined by combinatorial data:

Butti,ZaffaroniBenvenuti,Pando-Zayas,TachikawaLee,Rey

3i j k

, ,

9 Tr | V , V , V |32 i j k

i j k

a R a a a= = < >∑

12

d

ii

a=

=∑

Connection to a-maximization:

Thanks to a-maximization (Intriligator,Wecht) the exact R-charge of the CFT is obtained by maximizing a

There are various type of partition functions for BPS states:½ BPS states = Chiral Ring¼ BPS statesSupersymmetric Index

Counting problems in N=1 gauge theory

FIELD THEORY MOTIVATIONS:

Study of the moduli space: Number and structure of vacua

Dependence of the partition function on the coupling

Statistical properties of the BPS states and relation to black holes entropies

INTRINSIC MOTIVATIONS:

Deep interplay with various fields in math:Algebraic Geometry, Combinatorics, Invariant Theory

old and vast subject:

STRING MOTIVATIONS:

Count the number of BPS operators according to their global charge:

( )kn N = number of BPS operatorswith charge k

( ) ( ) kN kg a n N a=∑

•a is a chemical potential for global and R charges•N is the number of colors

Counting BPS states:Benvenuti,Feng,Hanany,He,Butti,Forcella,Vegh,A.Z.

Well studied problem in 2D: partition functions elliptic genus

Solvable in 4d for selected theories (duality BPS states – wrapped euclidean D3 branes)

contains information about:( )Ng a

•Density of states with dimension

•Structure of the moduli space: paramerized by gauge invariant operators dimension and number degrees of freedom

R charge∆ ∼

1

# . .( )N a d

d o fg aa→⎯⎯⎯→

Properties of BPS generating functions:Benvenuti,Feng,Hanany,He,Butti,Forcella,Vegh,A.Z.

Non conformal models and susy breaking:hanany,uranga,berenstein, bertolini, argurio,kachru, verlinde….

•Inclusion of fractional branes leads to cascadingnon conformal theories, with confining supersymmetricvacuum or runaway behavior.

•The runaway can be stabilized by embeddingin compact CY models

•Good laboratory for string realization of supersymmetrybreaking with metastable vacua

CONCLUSION

•The AdS/CFT correspondence is 10 years old

•One of major theoretical framework connecting string theoryto field theory (and sometimes math).

•It is still a continuous source of inspiration in field theory,model building and black hole entropy counting

•Holographic models for QCD, thermal and out of equilibriumphysics may still reserve interesting surprises