Integrability in Superconformal Chern-Simons Theories Konstantin Zarembo Ecole Normale Supérieure “Symposium on Theoretical and Mathematical Physics”, St. Petersburg, J.Minahan, K.Z., J.Minahan, W.Schulgin, K.Z., K.Z.,

Conformal theories At T = T c : CFT exact! Onsager’44 Belavin,Polyakov,Zamolodchikov’84 numerical Ising universality class:

Chern-Simons Abelian: Non-Abelian /SU(N)/: is an integer ( because of gauge invariance)

Particles interacting via Chern-Simons field: 2 1

1 2 linking number

2 1 Anyons Wilczek’82

Quantum Hall Effect Low-energy effective field theory for FQHE at filling fraction ν: Zhang,Hansson,Kivelson’89 - statistical gauge field

Chern-Simons-matter theories Not renormalizable: generated by RG Possible fixed points? Chen,Semenoff,Wu’92

How to find conformal points? Idea: use (super)symmetries. no relevant operators in the Lagrangian if marginal operators are related by symmetry to the CS term, their couplings do not run since k is not renormalized

Superconformal Chern-Simons D=3 Two gauge groups: Field content: in adjoint of in bifund. of

The Lagrangian Aharony,Bergman,Jafferis,Maldacena’08; Benna,Klebanov,Klose,Smedbäck’08; Hosomichi,Lee,Lee,Lee,Park’08

x1x1 x 3, …, x 10 Low-energy effective field theory of N multiple membranes in 10+1 dimensions x2x2 - transverse fluctuations (8 d.o.f.)

N=6 supersymmetry Conformal (k \in Z, no other adjustable couplings) Global symmetry: Symmetries Conformal group in 3d 10d rotations transverse to membrane

At, CP-invariant: if Level-rank duality: Enhanced suprsymmetry at k = 1 and 2 Aharony,Bergman,Jafferis’08 Non-perturbative dualities

Weak coupling Weak-coupling limit: ‘t Hooft expansion: small parameters: and

4D bulk 3D boundary z 0 Dual to string theory on AdS 4 x CP 3 AdS 4 : Aharony,Bergman,Jafferis,Maldacena’08

z 0 string propagator in the bulk Two-point correlation functions

AdS 4 /CFT 3 correspondence

Scaling dimensions In general, operators mix: anomalous dimension mixing matrix

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

cancel

Hamiltonian 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/ !

Integrable Hamiltonian - = Setting n→4 yields the CS mixing matrix! Standard construction of integrable Hamiltonian with su(4) symmetry: Leningrad school’70-80s

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:

Exact Diagonalization of many-body S-matrix Bethe equations Ahn,Nepomechie’08

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 Minahan,Sax,Sieg, to appear

Conclusions Planar N=6, D=3 Chern-Simons is integrable and solvable. Interpolating function h(λ)? θ-dependence? Q: 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? A: Yes, but very few, and only in D=2 and D=1 Z.’09