Integrability in Superconformal Chern-Simons Theories

Konstantin Zarembo

Ecole Normale Supérieure

“Symposium on Theoretical and Mathematical Physics”, St. Petersburg, 8.07.2009

J.Minahan, K.Z., 0806.3951

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

K.Z., 0903.1747

Conformal theories

At T = Tc: 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

AnyonsWilczek’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

x1

x 3 , …

, x10

Low-energy effective field theory

of N multiple membranes in 10+1 dimensions

x2

- 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 AdS4 x CP3

AdS4:

Aharony,Bergman,Jafferis,Maldacena’08

z

0

string propagator

in the bulk

Two-point correlation functions

AdS4/CFT3 correspondence

Scaling dimensions

In general, operators mix:

anomalous dimensionmixing matrix

^

Local operators and spin chains

i j

j i

Alternating spin chain of length 2L

^

cancel

Hamiltonian

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/ !

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 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:

Exact

Diagonalization of many-body S-matrix Bethe equations

Ahn,Nepomechie’08

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

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=1Z.’09