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.

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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:

Symmetries

Aharony,Bergman,Jafferis’08


Aharony,Bergman,Jafferis,Maldacena’08

AdS4/CFT3 correspondence


Local operators and spin chains

^

j

i

^

i

j

Alternating spin chain of length 2L


2

2

Mixing matrix

Minahan,Z.’08

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

  • 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


Exact solution

Gromov,Kazakov,Vieira’09

Y-system of thermodynamic Bethe ansatz:


Residual symmetries

Ground state:

Symmetry bearking:

Magnons:


Sigma-model in AdS4xCP3

φ

Z,Xa,X*a

Yi

t

CP3

AdS4


Light-cone gauge

Light-like geodesics:

gauge condition:


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


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

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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