the superconformal index for n=6 chern-simons theory

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”

• 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 Sd = 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 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


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


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: 2nd point will be useful.)


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 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

• 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


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 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.


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


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:

• ‘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)


S1 : Zk acts as translation

CP3

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:


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?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 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?13superconformal index for N=6 CS

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 ~ k14superconformal index for N=6 CS

Localization

• Index : path integral formulation in Euclidean QFT on S2 £ S1 .

• 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 (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

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


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


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

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.


or…

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 : 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


{4,3,3,2,1}

Numerical tests: 2 & 3 KK momenta

• Two KK momenta:


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


• Two KK momenta:


k = 2


• Two KK momenta:


k = 3


• Two KK momenta:

• Three KK momenta: k=1


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 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


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]


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


the superconformal index for n=6 chern-simons theory

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