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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
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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
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
U(1) x U(N2) case
U(2) x U(N2) caseq-hypergeometric function(q-ultraspherical function)
Schur Q-polynomial
double q-hypergeometricfunction
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)
The following integral representation renders the sum well-defined
regularized & analytically continued in the entire q-plane (“non-perturbative completion”)
P poles NP poles
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
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