The superconformal index for N=6 Chern-Simons theory

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					Indices for Superconformal Chern-Simons Theories

                       Seok Kim

                (Seoul National University)

                    9 December 2009
          “Current Trends in String Field Theory”
                     APCTP, Pohang.
                          Talk based on:

       S.K. , Nucl. Phys. B821, 241 (2009) [arXiv:0903.4172]
     S.K. and K. Madhu , JHEP 12 (2009) 018 [arXiv:0906.4751]

                             Related works:

 J. Bhattacharya and S. Minwalla, JHEP 0901, 014 (2009) [arXiv:0806.3251]
            M. Benna, I. Klebanov and T. Klose, arXiv:0906.3008
                 D. Berenstein and J. Park, arXiv:0906.3817
Y. Imamura and S. Yokoyama, Nucl.Phys. B827, 183 (2010) [arXiv:0908.0988]
• Chern-Simons-matter theories dual M-theory on AdS4              [Aharony-Bergman-
  Jafferis-Maldacena]…... (also [Bagger-Lambert] [Gustavsson]…)

• N=6 CSm with U(N)k x U(N)-k                  M-theory on AdS4 x S7 / Zk

                                                                  reduce on S1 / Zk

                                               IIA string theory on AdS4 x CP3

• Novelty: admits a microscopic study of M-theory

• A basic check: compare protected (or SUSY) local operators/states.

• More nontrivial than the same problem in, say, AdS5 /CFT4

• Why? Magnetic monopole operators: create magnetic flux on S2
  important for studying M-theory, SUSY enhancement, and so on.

                              Indices for Superconformal CS                           3
                    Introduction: monopole operators
• Local operators in R3 changing boundary conditions for fields: magnetic
   flux created on S2 surrounding the insertion point [„t Hooft] [Goddard-Nuyt-
   Olive] [Borokhov-Kapustin-Wu]

                                                               near r=0

• Above definition incomplete: should be supplemented by specifying
   behaviors of matter fields interacting with gauge fields

• Generically requires understanding on interaction: charge quantization

                               Indices for Superconformal CS                      4
                    Introduction: general problem
• Spectrum of gauge invariant operators involving monopoles

• Strategy: radial quantization, study states in CFT

• Question: energy (=dimension) & charges of states, degeneracy?

                           Indices for Superconformal CS           5
               Introduction: partition function & index
• These data are encoded in partition function: difficult to compute

• The index for supersymmetric CFT ( „superconformal index‟ ):

• In general, contains less information. Easier to compute

• Goal of this talk:

1. Explain how to compute the index including monopoles for N=6 CSm

2. Study properties of monopole operators

3. Discuss applications to other theories

                           Indices for Superconformal CS               6
                        Table of Contents

1.   Introduction

2.   The index for N=6 Chern-Simons theory

3.   Large N limit & a check of AdS4/CFT3

4.   Spectrum including monopoles for large CS level (optional)

5.   Concluding remarks

                        Indices for Superconformal CS             7
                N=6 Chern-Simons-matter theory
• N=6 Chern-Simons-matter theory [ABJM]: fields & charges   ( I=1,2,3,4 )

• Action

                         Indices for Superconformal CS                      8
            N=6 Chern-Simons-matter theory (continued)
• Moduli space of this theory is C4/Zk

• Large N dual to M-theory on AdS4 x S7/Zk : contains a circle S1/Zk

• Gauge theory dual of Kaluza-Klein momenta along this circle?

• The U(1)D in U(N) x U(N) is free: e.o.m. for

• Conserved U(1)b current: dual to M-theory momentum along S1 / Zk

                           Indices for Superconformal CS               9
                                    The index
• Superconformal symmetry of this theory is Osp(6|4).

• Important algebra: gives lower bound to energy (= D)

                                 SO(6) R-charge               SO(3) angular mom.
• Choose a pair Q & S: corresponding BPS states saturate the bound.

• Index counts states preserving Q,S.

• Charges weighting the states commute with Q, S : qi consists of
1.   (h1,h2) among the SO(6) Cartans (h1,h2,h3)
2.   The U(1)b charge h4

                              Indices for Superconformal CS                10
• One should take care of interaction. How?

• With (-1)F , one can add Q-exact operators inside trace:

• We work with the path integral: Euclidean QFT on S2 x S1 .

• This integral is “supersymmetric” with nilpotent Q

• One can add any Q-exact term to the action

• Large t limit: semi-classical „approximation‟ = exact result
                           Indices for Superconformal CS         11
                    Deformation and saddle points
• Use N=2 superfield formulation: auxiliary fields in vector multiplets

• Our choice: looks like d=3 „Yang-Mills‟ action (on S2 x S1 )

• saddle points: Dirac monopoles in U(1)N x U(1)N in U(N) x U(N) with
  holonomy along time circle.

   compactify r=1 & r=eβ

• Gaussian (1-loop) fluctuation (except holonomy)

                            Indices for Superconformal CS                 12
• „Classical action‟: 1-loop order

• 1-loop contribution: determinant over normal modes (= `letters‟)

• matter fields: monopole spherical harmonics, spectrum shifts
                                           for ij‟th bi-fundamental component

• Index over bi-fundamental modes:

•   adjoints (gauge fields, scalars, gauginos): indices mostly cancel

                           Indices for Superconformal CS                        13
                               Results (continued)
• Casimir „energy‟     (related quantity is also studied by [Benna-Klebanov-Klose])

• Should exactly integrate over holonomy

• This integral is the gauge singlet projection with Haar measure for
   „unbroken‟ gauge group. (E.g. H=(2,2,1,1,1,0) breaks U(6) to U(2) x U(3) x U(1) . )

• Index contribution from a saddle point:

                               Indices for Superconformal CS                          14
• Graviton index factorized:                               or

• Large N, low energy: most of the U(1)N do not support Dirac monopoles

• `Large‟ unbroken gauge groups U(N – O(1))

• identical holonomy variables          distributions [Brezin (‟78)]

• Integral over remaining holonomy factorizes

•                       was proven. [Bhattacharya-Minwalla]

• Nonperturbative in 1/k                : compare “D0 part” & “flux>0 part”
                           Indices for Superconformal CS                      15
                Sector with unit momentum (p11 = k/2)
• one saddle point: unit flux on both gauge groups

• Gauge theory result: integral over 2 holonomies

• Gravity: single graviton index in AdS4 x S7 projected to p11 = k/2

• One can show :

                           Indices for Superconformal CS               16
                        Sectors with higher momenta
• Many saddle points: e.g. with 2 fluxes, (2), (1,1) for each U(N)

• Flux distribution n along two U(1)N in U(N):

•             : used to study gauge invariant chiral operators (e.g., D = h3),
    SUSY enhancement, etc. [ABJM] [Berenstein-Trancanelli] [Hosomichi-Lee-Lee-Lee-
    Park-Yi] [Klebanov-Klose-Murugan] [Imamura] [Gaiotto-Jafferis] [Benna-Klebanov-Klose]
    [Gustavsson-Rey] [Berenstein-Park] [Kim-Madhu] , etc.

•             : gauge invariant BPS operators containing these monopoles
    appear in the index with nonzero angular momentum, D = h3 + j3 ( > h3 )

                                 Indices for Superconformal CS                        17
             Numerical tests: 2 & 3 KK momenta
• Two KK momenta: k = 1

• Three KK momenta: k=1

                           monopole operators with
          (for general k, appears from x2k+2 order higher than lowest order terms)
                           start from x2k+2 , x4k+2, x6k+6, resp., for general k

                          Indices for Superconformal CS                            18
              Direct analysis of spectrum for large k
• Want to understand the index from actual CSm theory [SK-Madhu]

• Origin of adjoint degrees and indices?

• Semi-classial approach: study linear excitations around classical
   monopole backgrounds, higher order interactions suppressed by 1/k

• We studied sectors with flux

• background solution: Gauss‟ law saturated by scalars in s-waves

                           Indices for Superconformal CS               19
                     Spectrum for large k (continued)
• Off-diagonal fluctuations „orthogonal‟ to the background ZI = (0,0,b1,b2):

                                        similar to modes in the deformed theory

• gauge fields & scalar fluctuation parallel to the background mix (see also
   [Berensetin-Park] )

• matter part of previous spectrum + a few more states in gauge fields

• some parallel scalar modes missing: constraints. Similar story for
                  E.g., bosonic constraints:

• Finite modes & constraints add up to adjoint index

                              Indices for Superconformal CS                       20
                             Concluding remarks
• Computed the index for N=6 CSm & compared with M-theory

• Various monopoles play crucial roles for M-theory vs. CS CFT3 duality

• Studied spectrum for large k: interaction with background monopole
   results in a finite number of adjoint degrees & constraints

• Study index for other CSm: check various non-perturbative proposals

• N=6,5 theories with U(N)k x U(N)-k or O(M)k x Sp(2N)-k [HLLLP] [ABJ]
 dualities between different field theories proposed

• N=4 theories:    [Gaiotto-Witten] [HLLLP], etc.

 Monopole index computed to test some ideas on gravity dual [Imamura-Yokoyama]

                               Indices for Superconformal CS               21
                        Concluding remarks (continued)
• N=3 with various gauge groups, matters [Jafferis-Tomasiello] [Hohenegger-
 Kirsch] [Gaiotto-Jafferis] [Hikida-Li-Takayanagi], etc.

 Seiberg duality [Giveon-Kutasov], [Niarchos], etc.

 Large N phase transition of the index [Bhattacharya-Bhattacharyya-Minwalla-Raju]

• Non-relativistic superconformal Chern-Simons-matter theories

• Conceptual questions on monopole operators:

1. What is the gauge theory dual of particle number in large N limit?
   E.g., low energy without monopoles: trace number ~ particle number in bulk

2. Study representations of the monopole operators in U(N) x U(N) via
   collective coordinate quantization [Benna-Klebanov-Klose] [Kapustin-Witten]
                                  Indices for Superconformal CS                  22

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