The superconformal index for N=6 Chern-Simons theory Seok Kim (Imperial College London) talk based on: arXiv:0903.4712 closely related works: J. Bhattacharya and S. Minwalla, JHEP 0901, 014 [arXiv:0806.3251]. F. Dolan, arXiv:0811.2740. J. Choi, S. Lee and J Song, JHEP 0903, 099 [arXiv:0811.2855]. Motivation • An important problem in AdS/CFT: study of the “spectrum” “operator-state map” : states in Sd = local (creation) operators at r=0 • energy (=scale dimension) & charges, degeneracy • Encoded in the partition function (if you can compute it…) superconformal index for N=6 CS 2 AdS/CFT and strong coupling • AdS/CFT often comes with coupling constants • Strong-weak duality: limited tools to study string theory & QFT ① CFT reliably studied in weakly-coupled regime ② SUGRA, -model… reliable at strong coupling • Spectrum acquires “large” renormalization: difficult to study • Examples: ① Yang-Mills coupling gYM , e.g. (N=4) Yang-Mills ② CS coupling k , e.g. (N=6) Chern-Simons-matter • This talk: some calculable strong coupling spectrum of N=6 CS superconformal index for N=6 CS 3 Supersymmetry • Supersymmetric CFT: energy bounded by conserved charges states preserving SUSY: saturate the bound • Supersymmetric Hilbert space: degeneracy. • Motivations to study supersymmetric states ① ② ③ ④ quantitative study of AdS/CFT supersymmetric black holes starting points for more elaborate studies (BMN, integrability, etc.) …… • SUSY partition function is still nontrivial: jump of SUSY spectrum superconformal index for N=6 CS 4 The Superconformal Index • States leave SUSY Hilbert space in boson-fermion pairs • The superconformal index counts #(boson) - #(fermion) . • “Witten index” + partition function : • Nice aspects: ① “topological” : index does not depend on continuous couplings ② Can use SUSY to compute it exactly at strongly coupled regime. (CS coupling k is discrete: 2nd point will be useful.) superconformal index for N=6 CS 5 Table of Contents 1. Motivation 2. Superconformal index for N=6 Chern-Simons theory 3. Outline of calculations 4. Testing AdS4/CFT3 for M-theory 5. Conclusion & Discussions superconformal index for N=6 CS 6 Superconformal algebra, BPS states & the Index • Superconformal algebra in d¸3 ① super-Poincare: P , J , Q ; conformal: D, K ; special SUSY S . ② R-symmetry Rij : U(N) or SO(2N) for N-extended SUSY in d=4,3 in radial quantization • Important algebra: gives lower bound to energy (= D) • For a given pair of Q & S, BPS states saturate this bound. • Index count states preserving Q,S . qi : charges commuting with Q, S superconformal index for N=6 CS 7 SCFT and indices in d=4 & d=3 • Index for d=4 SCFT: N=4 Yang-Mills ① does not depend on continuous gYM : compute in free theory ② agrees with index over gravitons in AdS5 x S5 • d=3 SCFT: Chern-Simons-matter theories, some w/ AdS4 M-theory duals [Bagger-Lambert] [Gustavsson] [Aharony-Bergman-Jafferis-Maldacena] ..... • Most supersymmetric: d=3, N=8 SUSY… • Next : N=6 theory with U(N)k x U(N)-k gauge group • (k,-k) Chern-Simons levels: discrete coupling. Index does depend on k. superconformal index for N=6 CS 8 N=6 Chern-Simons theory and the Index • N parallel M2’s near the tip of R8 / Zk : dual to M-theory on AdS4 x S7/Zk superconformal index for N=6 CS 9 N=6 Chern-Simons theory and the Index • N parallel M2’s near the tip of R8 / Zk : dual to M-theory on AdS4 x S7/Zk • Admits a type IIA limit for large k: S1 : Zk acts as translation CP3 • ‘t Hooft limit: large N keeping = N/k finite: ① weakly-coupled CS theory for small , IIA SUGRA, -model for large ② is effectively continuous [Bhattacharya-Minwalla] (caveat: energy is finite) superconformal index for N=6 CS 10 Index for free CS theory & type IIA SUGRA • dynamical fields: scalar CI (I=1,2,3,4), fermions I in • SUSY Q=Q1+i2- & S : SO(6)R to SO(2) x SO(4), BPS energy = q3 + J3 • ‘letters’ (operators made of single field) saturating BPS bound: • gauge invariants: • Free theory: no anomalous dimensions, count all of them. • 3 charges commute with Q,S: + J3 ; q1, q2 2 SO(4) . • Index: superconformal index for N=6 CS 11 Results (for type IIA) • Index over letters in & reps. (x = e- ) index over bi-fundamental index over anti-bi-fundamental • Full index : excite `identical’ letters & project to gauge singlets [Bose (Fermi) statistics] (also called ‘Plethystic exponential’) • graviton index: gravitons in AdS4 x S7 to zero KK momentum sector • Use large N technique: two indices agree [Bhattacharya-Minwalla] • Question: Can we study M-theory using the index? superconformal index for N=6 CS 12 Results (for type IIA) • Index over letters in & reps. (x = e- ) • Full index : excite `identical’ letters & project to gauge singlets • graviton index: gravitons in AdS4 x S7 to zero KK momentum sector • Use large N technique: two indices agree [Bhattacharya-Minwalla] • Question: Can we study M-theory using the index? superconformal index for N=6 CS 13 Gauge theory dual of M-theory states • M-theory states: carry KK momenta along fiber S1/Zk • Gauge theory dual [ABJM]: radially quantized theory on S2 x R • n flux : ( kn , -kn ) U(1) x U(1) electric charges induced. • Gauge invariant operators including magnetic monopole operators • No free theory limit with fluxes (flux quantization) • Finiteness of k crucial for studying M-theory states: p11 ~ k superconformal index for N=6 CS 14 Localization • Index : path integral formulation in Euclidean QFT on S2 £ S1 . • Path integral for index is supersymmetric with Q : localization 1. Nilpotent (Q2=0) symmetry: generated by translation by Grassmann number 2. Zero-mode ! volume factor: fermionic volume = 0 “Whole integral = 0” ??? 3. Caveat: There can be fixed points. Gaussian ‘approx.’ around fixed point = exact • More quantitative: One can insert any Q-exact term to the action • t!1 as semi-classical (Gaussian) ‘approximation’ superconformal index for N=6 CS 15 Calculation in N=6 Chern-Simons theory • Our choice: looks like d=3 ‘Yang-Mills’ action (on S^2 x S^1 ) superconformal index for N=6 CS 16 Calculation in N=6 Chern-Simons theory • Our choice: looks like d=3 ‘Yang-Mills’ action (on S^2 x S^1 ) • saddle points: Dirac monopoles in U(1)N x U(1)N of U(N) x U(N) with holonomy along time circle. • Gaussian (1-loop) fluctuation: ‘easily’ computable superconformal index for N=6 CS 17 Results (for M-theory) • Classical contribution: • charged fields: monopole spherical harmonics, letter indices shift • Indices for charged adjoints: gauge field & super-partners • Gauge invariance projection with unbroken gauge group Casimir energy superconformal index for N=6 CS 18 Tests • Gravity index is factorized as or… • Applying large N techniques, gauge theory index also factorizes • was proven. [Bhattacharya-Minwalla] • Nonperturbative: suffices to compare D0 brane part & flux>0 part. superconformal index for N=6 CS 19 Single D0 brane • 1 saddle point: unit flux on both gauge groups • Gauge theory result: • Gravity: single graviton index in AdS4 £ S7 ! project to p11 = k . • One can show : superconformal index for N=6 CS 20 Multi D0-branes • Flux distributions: With 2 fluxes, {2}, {1,1} for each U(1)N ½ U(N) • One can use Young diagrams for flux distributions: {4,3,3,2,1} • ‘Equal distributions’ : like or • monopole operators in conjugate representations of U(N) £ U(N) [ABJM] [Betenstein et.al.] [Klebanov et.al.] [Imamura] [Gaiotto et.al.]: easier to study • ‘Unequal distributions’ : like or • monopole operators in non-conjugate representations, unexplored superconformal index for N=6 CS 21 Numerical tests: 2 & 3 KK momenta • Two KK momenta: k = 1 monopole operators in non-conjugate representation of U(N) x U(N) chiral operators with 0 angular momentum [ABJM] [Hanany et.al.] [Berenstein et.al.] superconformal index for N=6 CS 22 Numerical tests: 2 & 3 KK momenta • Two KK momenta: k = 2 superconformal index for N=6 CS 23 Numerical tests: 2 & 3 KK momenta • Two KK momenta: k = 3 superconformal index for N=6 CS 24 Numerical tests: 2 & 3 KK momenta • Two KK momenta: k = 3 • Three KK momenta: k=1 superconformal index for N=6 CS 25 Conclusion & Discussions • Computed superconformal index for N=6 CS, compared with M-theory • Captures interacting spectrum: k dependence • Full set of monopole operators is very rich (e.g. non-conjugate rep.) • Crucial to understand M-theory / CS CFT3 duality • More to be done: 1. Direct understanding in physical Chern-Simons theory? [SK-Madhu] 2. Application to other Chern-Simons: e.g. test dualities using index superconformal index for N=6 CS 26 Conclusion and Discussions (continued) • N=5 theory with O(M)k x Sp(2N)-k [ABJ] [Hosomichi-Lee-Lee-Lee-Park] ‘Parity duality’ in CFT (strong-weak) : can be tested & studied by index • N=3 theories w/ fundamental matter [Giveon-Kutasov] [Gaiotto-Jafferis] etc. Seiberg duality, phase transition : study of flux sectors Implications to their gravity duals? • non-relativistic CS theory: monopole operators important [Lee-Lee-Lee] superconformal index for N=6 CS 27 Conclusion & Discussions (continued) • Last question: Any hint for N3/2 ? • In our case, degrees of freedom should scale as • Strong interaction should reduce d.o.f. by 1/2 . • Our index keeps some interactions superconformal index for N=6 CS 28

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