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Integrability in Superconformal Chern-Simons Theories

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.

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Integrability in Superconformal Chern-Simons Theories

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  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! Onsager’44 Belavin,Polyakov,Zamolodchikov’84

  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 Wilczek’82

  7. Quantum Hall Effect Low-energy effective field theory for FQHE at filling fraction ν: Zhang,Hansson,Kivelson’89 - statistical gauge field

  8. Chern-Simons-matter theories Not renormalizable: generated by RG Chen,Semenoff,Wu’92 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 • D=3 • Two gauge groups: • Field content: in adjoint of in bifund. of

  11. The Lagrangian Aharony,Bergman,Jafferis,Maldacena’08; Benna,Klebanov,Klose,Smedbäck’08; Hosomichi,Lee,Lee,Lee,Park’08

  12. x2 x1 x3 , … , x10 Low-energy effective field theory of N multiple membranes in10+1dimensions - transverse fluctuations(8 d.o.f.)

  13. N=6 supersymmetry Conformal (k \inZ, no other adjustable couplings) Global symmetry: Symmetries Conformal group in 3d 10d rotations transverse to membrane

  14. Non-perturbative dualities • At ,CP-invariant: • if • Level-rank duality: • Enhanced suprsymmetry at k = 1 and 2 Aharony,Bergman,Jafferis’08

  15. Weak coupling Weak-coupling limit: ‘t Hooft expansion: and small parameters:

  16. Dual to string theory onAdS4 x CP3 Aharony,Bergman,Jafferis,Maldacena’08 AdS4: z 4D bulk strings 0 Chern-Simons 3D boundary

  17. Two-point correlation functions z string propagator in the bulk 0

  18. AdS4/CFT3 correspondence

  19. Scaling dimensions In general, operators mix: mixing matrix anomalous dimension

  20. Local operators and spin chains ^ j i ^ i j Alternating spin chain of length 2L

  21. cancel

  22. 2 2 Hamiltonian Minahan,Z.’08 No dependence on Bak,Gang,Rey’08

  23. Integrability? Alternating SU(4) spin chain Integrable alternating spin chains /Faddeev,Reshetikhin’86/ generically involve next-to-nearest neighbour interactions /de Vega, Woynarovich’92/ !

  24. = - Integrable Hamiltonian Standard construction of integrable Hamiltonian with su(4) symmetry: Leningrad school’70-80s Setting n→4 yields the CS mixing matrix!

  25. Bethe ansatz equations Kulish,Reshetikhin’83 zero-momentum condition anomalous dimension

  26. Group theoretic Bethe equations Ogievetsky,Wiegmann’86 Cartan matrix: Dynkin labels of spin representation: (our case):

  27. 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

  28. All-loop asymptotic Bethe ansatz Gromov,Vieira’08 = dressing phase An unknown interpolating function for

  29. Exact solution Gromov,Kazakov,Vieira’09 Y-system of thermodynamic Bethe ansatz:

  30. Exact Ahn,Nepomechie’08 Diagonalization of many-body S-matrix Bethe equations

  31. Residual symmetries Ground state: Symmetry bearking: Magnons:

  32. Sigma-model in AdS4xCP3 φ Z,Xa,X*a Yi t CP3 AdS4

  33. Light-cone gauge Light-like geodesics: gauge condition:

  34. Sigma-model coupling constant: Classical limit is Setting t=τ=φ(light-cone gauge fixing) produces mass terms for transverse string fluctuations

  35. 8B+8F transverse oscillation modes, as required in critical superstring theory: Extra states, do not exist in the spin chain

  36. 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

  38. 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 Minahan ,Sax,Sieg, to appear

  39. 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 Conclusions Z.’09

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