Integrability in Superconformal Chern-Simons Theories
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Transcript of Integrability in Superconformal Chern-Simons Theories
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