ABJ Partition function Wilson Loops

and Seiberg Duality

with H. Awata, K. Nii (Nagoya U) & M. Shigemori (YITP)(1212.2966 & to appear soon)

KIAS Pre-Strings 2013

Shinji Hirano (University of the Witwatersrand)

ABJ(M) Conjecture Aharony-Bergman-Jefferis-(Maldacena)

M-theory on AdS4 x S7/Zk with (discrete) torsion C3

II

N=6 U(N1)k x U(N1+M)-k CSM theory

for large N1 and finite k

Discrete torsion

( fractional M2 = wrapped M5 )

IIA regime

large N1 and large k with λ = N1/k fixed

S7/Zk CP3 & C3 B2

Higher spin conjecture(Chang-Minwalla-Sharma-Yin)

N = 6 parity-violating Vasiliev’s higher spin theory

on AdS4

IIN = 6 U(N1)k x U(N2)-k CSM theory

with large N1 and k with fixed N1/k and finite N2

where

Why ABJ(M)? We are used to the idea

Localization of ABJ(M) theory

Classical Gravity

Strongly Coupled Gauge Theory @ large N

Strongly Coupled Gauge Theory @ 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 ( N1 = 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 AdS4 higher spin partition function

3. A study of Seiberg duality

In this talk1. 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 N2 - N2

Analytic continuation

perform all the eigenvalue integrals (Gaussian!)

U(N1) x U(N2) Lens space matrix model

ABJ Partition Function/Wilson loops

ABJ(M) Matrix Model• Localization yields (A = Φ = 0, D = - σ)

one-loop

where gs = -2πi/k

Lens space Matrix Model

Change of variables

VandermondeCosh Sinh

Gaussian integrals

Completely Gaussian!

N=N1+N2

multiple q-hypergeometricfunction

The lens space partition function

1. (q-Barnes G function)

(q-Gamma)

(q-number)

2. (q-Pochhammer)

U(1) x U(N2) case

U(2) x U(N2) caseq-hypergeometric function(q-ultraspherical function)

Schur Q-polynomial

double q-hypergeometricfunction

Analytic Continuation

Lens space MM ABJ MM

ABJ Partition FunctionU(N1) x U(N2) = U(N1) x U(N1+M) theory U(M) CS

Note: ZCS(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 poles NP poles

s

integration contour I

perturbative

non-perturbative

U(1)k x U(N)-k case (abelian Vasiliev on AdS4)

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 N1 < N2

for N1 < N2

1/2 BPS Wilson loop with winding n

s

integration contour I

perturbative

non-perturbative

Seiberg Duality

U(N1)k x U(N1+M)-k = U(N1+k-M)k x U(N1)-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

Discussions1. The Seiberg duality can be proven for

general N1 and N2

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