Transcript Bethe Ansatz in AdS/CFT: from local operators to classical

Integrability in Superconformal
Chern-Simons Theories
Konstantin Zarembo
Ecole Normale Supérieure
J.Minahan, K.Z., 0806.3951
J.Minahan, W.Schulgin, K.Z., 0901.1142
K.Z., 0903.1747
“Symposium on Theoretical and Mathematical Physics”, St. Petersburg, 8.07.2009
Conformal theories
CFT
At T = Tc:
Ising universality class:
numerical
exact! Onsager’44
Belavin,Polyakov,Zamolodchikov’84
Chern-Simons
Abelian:
Non-Abelian /SU(N)/:
is an integer ( because of gauge invariance)
Particles interacting via Chern-Simons field:
1
2
linking number
1
2
1
2
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
Chen,Semenoff,Wu’92
Possible fixed points?
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
x2
x1
Low-energy effective field theory
of N multiple membranes in 10+1 dimensions
- transverse fluctuations (8 d.o.f.)
Symmetries
• N=6 supersymmetry
• Conformal (k \in Z, no other adjustable couplings)
• Global symmetry:
Conformal group in 3d
10d rotations transverse to membrane
Non-perturbative dualities
• At
•
, CP-invariant:
if
• Level-rank duality:
Aharony,Bergman,Jafferis’08
• Enhanced suprsymmetry at k = 1 and 2
Weak coupling
Weak-coupling limit:
‘t Hooft expansion:
small parameters:
and
Dual to string theory on AdS4 x CP3
Aharony,Bergman,Jafferis,Maldacena’08
AdS4:
z
4D bulk
0
3D boundary
Two-point correlation functions
z
string propagator
in the bulk
0
AdS4/CFT3 correspondence
Scaling dimensions
In general, operators mix:
mixing matrix
anomalous dimension
Local operators and spin chains
^i
j
j
^i
Alternating spin chain of length 2L
cancel
Hamiltonian
Minahan,Z.’08
2
2
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
Standard construction of integrable Hamiltonian
Leningrad school’70-80s
with su(4) symmetry:
=
-
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
Checkedtranformation
for the single-fermion operators
of the Bethe equations
Minahan,Schulgin,Z.’09
Tsuboi’98
Beisert,Kazakov,Sakai,Z.’05
Zwiebel’09
Kazakov,Sorin,Zabrodin’07
 Consistent with supersymmetry
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
Ahn,Nepomechie’08
Diagonalization of many-body S-matrix
Bethe equations
Residual symmetries
Ground state:
Symmetry bearking:
Magnons:
Sigma-model in AdS4xCP3
φ
Z,Xa,X*a
Yi
t
3
CP
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
unless θ = 0 or π
/ex: O(3) sigma-model
Zamolodchikov,Zamolodchikov’92
/
θ-dependence at weak coupling:
cancels at two loops Bak,Gang,Rey’08; Zwiebel’09; Minahan,Schulgin,Z.’09
four loops? 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