The superconformal index for N=6 Chern-Simons theory Seok Kim (Imperial College London) talk based on: arXiv: closely related works: J. Bhattacharya and S. Minwalla, JHEP 0901, 014 [arXiv: ]. F. Dolan, arXiv: J. Choi, S. Lee and J Song, JHEP 0903, 099 [arXiv: ].

Motivation An important problem in AdS/CFT: study of the “spectrum” energy (=scale dimension) & charges, degeneracy Encoded in the partition function (if you can compute it…) 2superconformal index for N=6 CS “operator-state map” : states in S d = local (creation) operators at r=0

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 g YM, 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 3superconformal index for N=6 CS

Supersymmetry Supersymmetric CFT: energy bounded by conserved charges Supersymmetric Hilbert space: degeneracy. Motivations to study supersymmetric states ① quantitative study of AdS/CFT ② supersymmetric black holes ③ starting points for more elaborate studies (BMN, integrability, etc.) ④ …… SUSY partition function is still nontrivial: jump of SUSY spectrum 4superconformal index for N=6 CS states preserving SUSY: saturate the bound

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: 2 nd point will be useful.) 5superconformal index for N=6 CS

Table of Contents 1.Motivation 2.Superconformal index for N=6 Chern-Simons theory 3.Outline of calculations 4.Testing AdS 4 /CFT 3 for M-theory 5.Conclusion & Discussions 6superconformal index for N=6 CS

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 R ij : U(N) or SO(2N) for N-extended SUSY in d=4,3 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. q i : charges commuting with Q, S 7superconformal index for N=6 CS in radial quantization

SCFT and indices in d=4 & d=3 Index for d=4 SCFT: N=4 Yang-Mills ① does not depend on continuous g YM : compute in free theory ② agrees with index over gravitons in AdS 5 x S 5 d=3 SCFT: Chern-Simons-matter theories, some w/ AdS 4 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. 8superconformal index for N=6 CS

N=6 Chern-Simons theory and the Index N parallel M2’s near the tip of R 8 / Z k : dual to M-theory on AdS 4 x S 7 /Z k 9superconformal index for N=6 CS

N=6 Chern-Simons theory and the Index N parallel M2’s near the tip of R 8 / Z k : dual to M-theory on AdS 4 x S 7 /Z k Admits a type IIA limit for large k: ‘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) 10superconformal index for N=6 CS S 1 : Z k acts as translation CP 3

Index for free CS theory & type IIA SUGRA dynamical fields: scalar C I (I=1,2,3,4), fermions  I  in SUSY Q=Q 1+i2 - & S : SO(6) R to SO(2) x SO(4), BPS energy  = q 3 + J 3 ‘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:  + J 3 ; q 1, q 2 2 SO(4). Index: 11superconformal index for N=6 CS

Results (for type IIA) Index over letters in & reps. (x = e -  ) Full index : excite `identical’ letters & project to gauge singlets graviton index: gravitons in AdS 4 x S 7 to zero KK momentum sector Use large N technique: two indices agree [Bhattacharya-Minwalla] Question: Can we study M-theory using the index? 12superconformal index for N=6 CS index over bi-fundamental index over anti-bi-fundamental [Bose (Fermi) statistics] (also called ‘Plethystic exponential’)

Results (for type IIA) Index over letters in & reps. (x = e -  ) Full index : excite `identical’ letters & project to gauge singlets graviton index: gravitons in AdS 4 x S 7 to zero KK momentum sector Use large N technique: two indices agree [Bhattacharya-Minwalla] Question: Can we study M-theory using the index? 13superconformal index for N=6 CS

Gauge theory dual of M-theory states M-theory states: carry KK momenta along fiber S 1 /Z k Gauge theory dual [ABJM] : radially quantized theory on S 2 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: p 11 ~ k 14superconformal index for N=6 CS

Localization Index : path integral formulation in Euclidean QFT on S 2 £ S 1. Path integral for index is supersymmetric with Q : localization More quantitative: One can insert any Q-exact term to the action t!1 as semi-classical (Gaussian) ‘approximation’ 15superconformal index for N=6 CS 1. Nilpotent (Q 2 =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

Calculation in N=6 Chern-Simons theory Our choice: looks like d=3 ‘Yang-Mills’ action (on S^2 x S^1 ) 16superconformal index for N=6 CS

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 17superconformal index for N=6 CS

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 18superconformal index for N=6 CS Casimir energy

Tests Gravity index is factorized as Applying large N techniques, gauge theory index also factorizes was proven. [Bhattacharya-Minwalla] Nonperturbative: suffices to compare D0 brane part & flux>0 part. 19superconformal index for N=6 CS or…

Single D0 brane 1 saddle point: unit flux on both gauge groups Gauge theory result: Gravity: single graviton index in AdS 4 £ S 7 ! project to p 11 = k. One can show : 20superconformal index for N=6 CS

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: ‘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 21superconformal index for N=6 CS {4,3,3,2,1}

Numerical tests: 2 & 3 KK momenta Two KK momenta: 22superconformal index for N=6 CS chiral operators with 0 angular momentum [ABJM] [Hanany et.al.] [Berenstein et.al.] monopole operators in non-conjugate representation of U(N) x U(N) k = 1

Numerical tests: 2 & 3 KK momenta Two KK momenta: 23superconformal index for N=6 CS k = 2

Numerical tests: 2 & 3 KK momenta Two KK momenta: 24superconformal index for N=6 CS k = 3

Numerical tests: 2 & 3 KK momenta Two KK momenta: Three KK momenta: k=1 25superconformal index for N=6 CS k = 3

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 CFT 3 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 26superconformal index for N=6 CS

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] 27superconformal index for N=6 CS

Conclusion & Discussions (continued) Last question: Any hint for N 3/2 ? In our case, degrees of freedom should scale as Strong interaction should reduce d.o.f. by 1/2. Our index keeps some interactions 28superconformal index for N=6 CS