Integrability and AdS/CFT correspondence in three dimensions Konstantin Zarembo École Normale Supérieure Paris “Sakharov Conference”, Moscow, J.Minahan, K.Z., J.Minahan, W.Schulgin, K.Z., K.Z., and in progress

AdS/CFT correspondence Yang-Mills theory with N=4 supersymmetry String theory on AdS 5 xS 5 background Maldacena’97 Gubser,Klebanov,Polyakov’98 Witten’98 N=6 Supersymmetric Chern-Simons-matter theory String theory on AdS 4 xCP 3 background Aharony,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 Z 4 symmetric G/H 0 coset: g – coset representative: String sigma-model: Serganova’83 Metsaev,Tseytlin’98 Roiban,Siegel’00 BB F F

1. Integrable follows from Z 4 symmetry Bena,Polchinski,Roiban’03 2. Conformal (β-function = 0) 3. Central charge = 26 Super AdS 5 x S 5 Super AdS 4 x CP 3 Z., in progress

Superconformal Chern-Simons D=3 (dual to AdS 4 x CP 3 ) 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

AdS 4 /CFT 3 correspondence Aharony,Bergman,Jafferis,Maldacena’08

^ Local operators and spin chains i j j i Alternating spin chain of length 2L ^

Mixing matrix Minahan,Z.’ 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 equations Ogievetsky,Wiegmann’86 Cartan matrix: Dynkin labels of spin representation: (our case):

Full spectrum Duality tranformation of the Bethe equations Tsuboi’98 Beisert,Kazakov,Sakai,Z.’05 Kazakov,Sorin,Zabrodin’07 Checked for the single-fermion operators Consistent with supersymmetry Minahan,Schulgin,Z.’09 Zwiebel’09

All-loop asymptotic Bethe ansatz Gromov,Vieira’08 = dressing phase An unknown interpolating function for

Exact solution Gromov,Kazakov,Vieira’09 Y-system of thermodynamic Bethe ansatz:

Residual symmetries Ground state: Symmetry bearking: Magnons:

φ Z,X a,X * a t YiYi CP 3 AdS 4 Sigma-model in AdS 4 xCP 3

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 loops four 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?