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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] Introduction • 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 Localization • 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 Results • „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: where • 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 Tests • 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 et.al. (‟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 2 3 • 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 fermions 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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