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Integrability and AdS/CFT correspondence in three dimensions

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### Integrability and AdS/CFT correspondence in three dimensions

Konstantin Zarembo

École Normale Supérieure

Paris

J.Minahan, K.Z., 0806.3951

J.Minahan, W.Schulgin, K.Z., 0901.1142

K.Z., 0903.1747 and in progress

“Sakharov Conference”, Moscow, 18.05.2009

AdS/CFT correspondence

D=4

String theory on

AdS5xS5 background

Yang-Mills theory

with N=4 supersymmetry

Maldacena’97

Gubser,Klebanov,Polyakov’98

Witten’98

D=3

String theory on

AdS4xCP3 background

N=6 Supersymmetric

Chern-Simons-matter theory

Aharony,Bergman,Jafferis,Maldacena’08

Aharony,Bergman,Jafferis’08

these two cases are unique in certain sense

Z., to appear

Semi-symmetric superspaces

Serganova’83

Z4symmetric G/H0 coset:

B

B

F

F

g – coset representative:

String sigma-model:

Metsaev,Tseytlin’98

Roiban,Siegel’00

1. Integrable

follows fromZ4symmetry

Bena,Polchinski,Roiban’03

2. Conformal (β-function = 0)

Z., in progress

3. Central charge = 26

Super AdS4 x CP3

Super AdS5 x S5

Superconformal Chern-Simons

- D=3 (dual to AdS4x CP3)
- Two gauge groups:
- Field content:

in adjoint of

in bifund. of

spinor index of SO(6) R-symmetry

The Lagrangian

Aharony,Bergman,Jafferis,Maldacena’08;

Benna,Klebanov,Klose,Smedbäck’08;

Hosomichi,Lee,Lee,Lee,Park’08

N=6 supersymmetry

Conformal (k is integer – cannot be renormalized)

Global symmetry:

Large-N limit:

‘t Hooft couplings:

At ,CP-invariant:

Non-perturbative dualities:

if

level-rank duality:

SymmetriesAharony,Bergman,Jafferis’08

Aharony,Bergman,Jafferis,Maldacena’08

AdS4/CFT3 correspondence

Integrability?

Alternating SU(4) spin chain

Integrable alternating spin chains /Faddeev,Reshetikhin’86/ generically

involve next-to-nearest neighbour interactions /de Vega, Woynarovich’92/ !

Group theoretic Bethe equations

Ogievetsky,Wiegmann’86

Cartan matrix:

Dynkin labels of spin representation:

(our case):

Full spectrum

Duality tranformation

of the Bethe equations

- Checked for the single-fermion operators
- Consistent with supersymmetry

Minahan,Schulgin,Z.’09

Tsuboi’98

Beisert,Kazakov,Sakai,Z.’05

Kazakov,Sorin,Zabrodin’07

Zwiebel’09

All-loop asymptotic Bethe ansatz

Gromov,Vieira’08

= dressing phase

An unknown interpolating function

for

Sigma-model coupling constant:

Classical limit

is

Setting t=τ=φ(light-cone gauge fixing) produces mass

terms for transverse string fluctuations

8B+8F transverse oscillation modes,

as required in critical superstring theory:

Extra states,

do not exist in the spin chain

Propagator of the heavy mode:

Near threshold the one-loop correction cannot be neglected:

pole disappears

heavy string modes dissolve

in the two-particle continuum

of light modes

θ-dependence

Folklore: sigma-models cannot be integrable

unlessθ = 0 or π

/ex: O(3) sigma-modelZamolodchikov,Zamolodchikov’92/

- θ-dependence at weak coupling:
- cancels at two loops
- four loops?

Bak,Gang,Rey’08; Zwiebel’09; Minahan,Schulgin,Z.’09

Planar N=6, D=3 Chern-Simons is integrable and solvable.

Interpolating function h(λ)?

θ-dependence?

Are there other integrable/solvable large-N CFTs, apart from N=4, D=4 super-Yang-Mills and N=6, D=3 super-Chern-Simons?

Conclusions
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