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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

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
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
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
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
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:

Symmetries

Aharony,Bergman,Jafferis’08


Aharony,Bergman,Jafferis,Maldacena’08

AdS4/CFT3 correspondence


Local operators and spin chains
Local operators and spin chains

^

j

i

^

i

j

Alternating spin chain of length 2L


Mixing matrix

2

2

Mixing matrix

Minahan,Z.’08

No dependence on

Bak,Gang,Rey’08


Integrability
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

=

=

R-matrices

Monodromy matrices:


Yang baxter equation
Yang-Baxter equation

Extra YBE:

only if


Integrable hamiltonian

=

-

Integrable Hamiltonian

Transfer- matrices:

Hamiltonians:

Setting n→4 yields the CS mixing matrix!


Bethe ansatz equations
Bethe ansatz equations

Kulish,Reshetikhin’83

zero-momentum condition

anomalous dimension


Group theoretic bethe equations
Group theoretic Bethe equations

Ogievetsky,Wiegmann’86

Cartan matrix:

Dynkin labels of spin representation:

(our case):


Full spectrum
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
All-loop asymptotic Bethe ansatz

Gromov,Vieira’08

= dressing phase

An unknown interpolating function

for


Exact solution
Exact solution

Gromov,Kazakov,Vieira’09

Y-system of thermodynamic Bethe ansatz:


Residual symmetries
Residual symmetries

Ground state:

Symmetry bearking:

Magnons:


Sigma model in ads 4 xcp 3
Sigma-model in AdS4xCP3

φ

Z,Xa,X*a

Yi

t

CP3

AdS4


Light cone gauge
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



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


Dependence
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


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?

Conclusions


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