Integrability and AdS/CFT correspondence in three dimensions

Konstantin Zarembo

École Normale Supérieure

Paris

“Sakharov Conference”, Moscow, 18.05.2009

J.Minahan, K.Z., 0806.3951

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

K.Z., 0903.1747 and in progress

AdS/CFT correspondence

Yang-Mills theory

with N=4 supersymmetry

String theory on

AdS5xS5 backgroundMaldacena’97

Gubser,Klebanov,Polyakov’98

Witten’98

N=6 Supersymmetric

Chern-Simons-matter theory

String theory on

AdS4xCP3 backgroundAharony,Bergman,Jafferis,Maldacena’08

Aharony,Bergman,Jafferis’08

D=4

D=3

these two cases are unique in certain sense Z., to appear

Semi-symmetric superspaces

Z4 symmetric G/H0 coset:

g – coset representative:

String sigma-model:

Serganova’83

Metsaev,Tseytlin’98

Roiban,Siegel’00

BB

F F

1. Integrable follows from Z4 symmetryBena,Polchinski,Roiban’03

2. Conformal (β-function = 0)

3. Central charge = 26

Super AdS5 x S5 Super AdS4 x CP3

Z., in progress

Superconformal Chern-Simons

• D=3 (dual to AdS4 x 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

Symmetries

• 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:

Aharony,Bergman,Jafferis’08

AdS4/CFT3 correspondenceAharony,Bergman,Jafferis,Maldacena’08

^

Local operators and spin chains

i j

j i

Alternating spin chain of length 2L

^

Mixing matrix

Minahan,Z.’08

22

No dependence on Bak,Gang,Rey’08

Integrability?

Alternating SU(4) spin chain

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

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

R-matrices

Monodromy matrices:

=

=

Yang-Baxter equation

Extra YBE:

only if

Integrable Hamiltonian

Transfer- matrices:

Hamiltonians:

-=

Setting n→4 yields the CS mixing matrix!

Bethe ansatz equations

Kulish,Reshetikhin’83

zero-momentum condition

anomalous dimension

Group theoretic Bethe equationsOgievetsky,Wiegmann’86

Cartan matrix:

Dynkin labels of spin representation:

(our case):

Full spectrum

Duality tranformation

of the Bethe equationsTsuboi’98

Beisert,Kazakov,Sakai,Z.’05

Kazakov,Sorin,Zabrodin’07

Checked for the single-fermion operators

Consistent with supersymmetryMinahan,Schulgin,Z.’09

Zwiebel’09

All-loop asymptotic Bethe ansatzGromov,Vieira’08

= dressing phase

An unknown interpolating function for

Exact solutionGromov,Kazakov,Vieira’09

Y-system of thermodynamic Bethe ansatz:

Residual symmetries

Ground state:

Symmetry bearking:

Magnons:

φZ,Xa,X*a

t

Yi

CP3 AdS4

Sigma-model in AdS4xCP3

Light-cone gauge

Light-like geodesics:

gauge condition:

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

terms for transverse string fluctuations

Sigma-model coupling constant:Classical limit

is

8B+8F transverse oscillation modes,

as required in critical superstring theory:

Extra states,

do not exist in the spin chain

Worldsheet interactions

Z.’09

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

unless θ = 0 or π

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

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

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

Conclusions

• 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?