Recent developments in AdS/CFT correspondence

Kostya Zarembo

École Normale Supérieure

&

Uppsala U.

Rencontres de Moriond, La Thuile, 10.03.2008

Yang-Mills fields = closed strings

“rings of glue”

Polyakov’80

or maybe ≈

Planar diagrams and strings

time

Large-N limit: ‘t Hooft’74

Conceptual problems:

• Closed strings describe gravity. What is graviton in YM?

• String theory is only consistent in ten dimensions.

• How does the string remember that it is made of gluons?

Resolves many puzzles of putative large-N string:

• graviton is not a massless glueball, but is the dual of the energy-momentum tensor Tμν

• extra dimensions are geometric images of the energy scale and of global symmetries

• sum rules are automatic

String statesBound states in QFT

(mesons, glueballs)

String states Local operators

AdS/CFT correspondence

Yang-Mills theory with

N=4 supersymmetry

String theory on

AdS5xS5 background

Maldacena’97

Gubser,Klebanov,Polyakov’98

Witten’98Exact equivalence

Anti-de-Sitter space (AdS5)

5D bulk

4D boundary

z

0

AdS/CFT correspondence Maldacena’97

Gubser,Klebanov,Polyakov’98

Witten’98

Strong-weak coupling interpolation

Circular Wilson loop (exact):Erickson,Semenoff,Zarembo’00

Drukker,Gross’00

Pestun’07

0 λSYM perturbation

theory

1 + + …+

String perturbation

theory

Minimal area law in AdS5

Correlation functions

Dilatation operator:

matrix of anomalous dimensions

Local operators and spin chains

i j

i j

One loop planar mixing matrix:

Minahan,Z.’02

Heisenberg Hamiltonian

The spectrum

Ground state:

Excited states (magnons):

Exact periodicity condition:

momentumscattering phase shifts

periodicity of wave function

Anomalous dimension:

Bethe’31

Bethe equations for Heisenberg model

Rapidity:

Full asymptotic BABeisert,Staudacher’05

Wrapping/finite size effects

Ambjørn,Janik,Kristjansen’05

Weak coupling:

wrapping ordercaptured by

ABABeisert,Kristjansen,Staudacher’03

Large-spin twist-2 operators

where

Beisert,Eden,Staudacher’06

Weak coupling:

Beisert,Eden,Staudacher’06

Confirmed by explicit four-loop calculations in SYMBern,Czakon,Dixon,Kosower,Smirnov’06; Cachazo,Spradlin,Volovich’06

Strong coupling:

Basso,Korchemsky,Kotański’07

Confirmed by explicit two-loop calculations in

string theory Roiban,Tseytlin’07; Klose,Minahan,McLoughlin,Z.’07

Conclusions

• Large-N SYM is a solvable (and to certain degree solved) 4d theory. The solution is not simple!

• Finite-size/wrapping effects (Asymptotic Bethe Ansatz => Thermodynamic Bethe Ansatz?)

• String theory <=> Yang-Mills <=> spin chain

How does it work in detail? • What can we learn from solving AdS/CFT exactly?