Integrability in Superconformal Chern-Simons Theories

1
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
2
Conformal theories
CFT
At T Tc
Ising universality class
numerical
exact!
Onsager44 Belavin,Polyakov,Zamolodchikov84
3
Chern-Simons
Abelian
Non-Abelian /SU(N)/
is an integer ( because of gauge invariance)
4
Particles interacting via Chern-Simons field
1
2
5
linking number
2
1
6
1
2
Anyons
Wilczek82
7
Quantum Hall Effect
Low-energy effective field theory for FQHE at
filling fraction ?
Zhang,Hansson,Kivelson89
- statistical gauge field
8
Chern-Simons-matter theories
Not renormalizable
generated by RG
Chen,Semenoff,Wu92
Possible fixed points?
9
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

10
Superconformal Chern-Simons

• D3
• Two gauge groups
• Field content

in adjoint of
in bifund. of
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The Lagrangian
Aharony,Bergman,Jafferis,Maldacena08
Benna,Klebanov,Klose,Smedbäck08
Hosomichi,Lee,Lee,Lee,Park08
12
x2
x1
x3 , , x10
Low-energy effective field theory of N multiple
membranes in 101 dimensions
- transverse fluctuations (8 d.o.f.)
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Symmetries

• N6 supersymmetry
• Conformal (k \in Z, no other adjustable
  couplings)
• Global symmetry

Conformal group in 3d
10d rotations transverse to membrane
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Non-perturbative dualities

• At , CP-invariant
• if
• Level-rank duality
• Enhanced suprsymmetry at k 1 and 2

Aharony,Bergman,Jafferis08
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Weak coupling
Weak-coupling limit
t Hooft expansion
and
small parameters
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Dual to string theory on AdS4 x CP3
Aharony,Bergman,Jafferis,Maldacena08
AdS4
z
4D bulk
strings
0
Chern-Simons
3D boundary
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Two-point correlation functions
z
string propagator in the bulk
0
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AdS4/CFT3 correspondence
19
Scaling dimensions
In general, operators mix
mixing matrix
anomalous dimension
20
Local operators and spin chains

i
j

j
i
Alternating spin chain of length 2L
21
cancel
22
Hamiltonian
Minahan,Z.08
No dependence on
Bak,Gang,Rey08
23
Integrability?
Alternating SU(4) spin chain
Integrable alternating spin chains
/Faddeev,Reshetikhin86/ generically involve
next-to-nearest neighbour interactions /de Vega,
Woynarovich92/ !
24
Integrable Hamiltonian
Standard construction of integrable
Hamiltonian with su(4) symmetry
Leningrad school70-80s
Setting n?4 yields the CS mixing matrix!
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Bethe ansatz equations
Kulish,Reshetikhin83
zero-momentum condition
anomalous dimension
26
Group theoretic Bethe equations
Ogievetsky,Wiegmann86
Cartan matrix
Dynkin labels of spin representation
(our case)
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Full spectrum
Duality tranformation of the Bethe equations

• Checked for the single-fermion operators
• Consistent with supersymmetry

Minahan,Schulgin,Z.09
Tsuboi98 Beisert,Kazakov,Sakai,Z.05 Kazakov,Sori
n,Zabrodin07
Zwiebel09
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All-loop asymptotic Bethe ansatz
Gromov,Vieira08
dressing phase
An unknown interpolating function
for
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Exact solution
Gromov,Kazakov,Vieira09
Y-system of thermodynamic Bethe ansatz
30
Exact
Ahn,Nepomechie08
Diagonalization of many-body S-matrix
Bethe equations
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Residual symmetries
Ground state
Symmetry bearking
Magnons
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Sigma-model in AdS4xCP3
f
Z,Xa,Xa
Yi
t
CP3
AdS4
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Light-cone gauge
Light-like geodesics
gauge condition
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Sigma-model coupling constant
Classical limit is
Setting ttf (light-cone gauge fixing) produces
mass terms for transverse string fluctuations
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8B8F transverse oscillation modes, as required
in critical superstring theory
Extra states, do not exist in the spin chain
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Worldsheet interactions
Z.09
37
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
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?-dependence
Folklore sigma-models cannot be integrable
unless ? 0 or p
/ex O(3) sigma-model Zamolodchikov,Zamolodchikov
92/

• ?-dependence at weak coupling
• cancels at two loops
• four loops?

Bak,Gang,Rey08 Zwiebel09 Minahan,Schulgin,Z.0
9
Minahan ,Sax,Sieg, to appear
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Conclusions

• Planar N6, D3 Chern-Simons is integrable and
  solvable.
• Interpolating function h(?)?
• ?-dependence?
• Q Are there other integrable/solvable large-N
  CFTs, apart from N4, D4 super-Yang-Mills and
  N6, D3 super-Chern-Simons?
  A Yes, but very few, and
  only in D2 and D1

Z.09