with H. Awata, K. Nii (Nagoya U) & M. Shigemori (YITP) ( & to appear soon) KIAS Pre-Strings 2013 Shinji Hirano (University of the Witwatersrand)

ABJ(M) Conjecture Aharony-Bergman-Jefferis-(Maldacena) M-theory on AdS 4 x S 7 /Z k with (discrete) torsion C 3 II N=6 U(N 1 ) k x U(N 1 +M) -k CSM theory for large N 1 and finite k

 Discrete torsion ( fractional M2 = wrapped M5 )  IIA regime large N 1 and large k with λ = N 1 /k fixed S 7 /Z k  CP 3 & C 3  B 2

Higher spin conjecture (Chang-Minwalla-Sharma-Yin) N = 6 parity-violating Vasiliev’s higher spin theory on AdS 4 II N = 6 U(N 1 ) k x U(N 2 ) -k CSM theory with large N 1 and k with fixed N 1 /k and finite N 2 where

Why ABJ(M)?  We are used to the idea  Localization of ABJ(M) theory Classical Gravity Strongly Coupled Gauge large N Strongly Coupled Gauge finite N “Quantum Gravity” Integrability goes both ways and deals with non-BPS but large N Localization goes this way and deals only with BPS but finite N

Progress to date  The ABJM partition function ( N 1 = N, M = 0 ) Perturbative “Quantum Gravity” Partition Function II Airy Function A mismatch in 1/N correction AdS radius shift Leading

Why ABJ? 1. Does Airy persist with the AdS radius shift with B field ? (presumably yes) 2. A prediction on the AdS 4 higher spin partition function 3. A study of Seiberg duality

In this talk 1. Study ABJ partition function & Wilson loops and their behaviors under Seiberg duality 2. Do not answer Q1 & Q2 but make progress to the point that these answers are within the reach 3. Answer Q3 with reasonable satisfaction

ABJ Partition Function

Our Strategy rank N 2  - N 2 Analytic continuation perform all the eigenvalue integrals (Gaussian!) U(N 1 ) x U(N 2 ) Lens space matrix model ABJ Partition Function/Wilson loops

ABJ(M) Matrix Model Localization yields (A = Φ = 0, D = - σ) one-loop where g s = -2πi/k

Lens space Matrix Model

Change of variables Vandermonde Cosh  Sinh

Gaussian integrals Completely Gaussian! N=N 1 +N 2

multiple q-hypergeometric function The lens space partition function

1. (q-Barnes G function) (q-Gamma) (q-number) 2. (q-Pochhammer)

 U(1) x U(N 2 ) case  U(2) x U(N 2 ) case q-hypergeometric function (q-ultraspherical function) Schur Q-polynomial double q-hypergeometric function

Analytic Continuation Lens space MM  ABJ MM

ABJ Partition Function U(N 1 ) x U(N 2 ) = U(N 1 ) x U(N 1 +M) theory U(M) CS Note: Z CS (M) k = 0 for M > k (SUSY breaking)

Integral Representation  The sum is a formal series not convergent, not well-defined at for even k

 The following integral representation renders the sum well-defined regularized & analytically continued in the entire q-plane (“non-perturbative completion”) P polesNP poles

s integration contour I perturbative non-perturbative

 U(1) k x U(N) -k case (abelian Vasiliev on AdS 4 ) This is simple enough to study the higher spin limit

ABJ Wilson Loops

 1/6 BPS Wilson loops with winding n

Wilson loop results for N 1 < N 2

 1/2 BPS Wilson loop with winding n

s integration contour I perturbative non-perturbative

Seiberg Duality

U(N 1 ) k x U(N 1 +M) -k = U(N 1 +k-M) k x U(N 1 ) -k

Partition function (Example)

 The partition functions of the dual pair More generally

Fundamental Wilson loops  1/6 BPS Wilson loops  1/2 BPS Wilson loops

Discussions 1. The Seiberg duality can be proven for general N 1 and N 2 2. Wilson loops in general representations 3. The Fermi gas approach to the ABJ theory (non-interacting & only simple change in the density matrix) 4. Interesting to study the transition from higher spin fields to strings

The End