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KIAS Pre-Strings 2013. w ith H. Awata , K . Nii (Nagoya U) & M. Shigemori (YITP) (1212.2966 & to appear soon). ABJ Partition function Wilson Loops and Seiberg Duality. Shinji Hirano ( University of the Witwatersrand ). ABJ(M) Conjecture Aharony -Bergman- Jefferis -( Maldacena ). - PowerPoint PPT Presentation
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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
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