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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)
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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
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Discrete torsion
( fractional M2 = wrapped M5 )
IIA regime
large N1 and large k with λ = N1/k fixed
S7/Zk CP3 & C3 B2
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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
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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
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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
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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
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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
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ABJ Partition Function
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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
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ABJ(M) Matrix Model• Localization yields (A = Φ = 0, D = - σ)
one-loop
where gs = -2πi/k
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Lens space Matrix Model
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Change of variables
VandermondeCosh Sinh
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Gaussian integrals
Completely Gaussian!
N=N1+N2
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multiple q-hypergeometricfunction
The lens space partition function
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1. (q-Barnes G function)
(q-Gamma)
(q-number)
2. (q-Pochhammer)
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U(1) x U(N2) case
U(2) x U(N2) caseq-hypergeometric function(q-ultraspherical function)
Schur Q-polynomial
double q-hypergeometricfunction
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Analytic Continuation
Lens space MM ABJ MM
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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)
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Integral Representation The sum is a formal series
not convergent, not well-defined at for even k
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The following integral representation renders the sum well-defined
regularized & analytically continued in the entire q-plane (“non-perturbative completion”)
P poles NP poles
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s
integration contour I
perturbative
non-perturbative
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U(1)k x U(N)-k case (abelian Vasiliev on AdS4)
This is simple enough to study the higher spin limit
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ABJ Wilson Loops
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1/6 BPS Wilson loops with winding n
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Wilson loop results
for N1 < N2
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for N1 < N2
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1/2 BPS Wilson loop with winding n
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s
integration contour I
perturbative
non-perturbative
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Seiberg Duality
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U(N1)k x U(N1+M)-k = U(N1+k-M)k x U(N1)-k
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Partition function (Example)
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The partition functions of the dual pair
More generally
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Fundamental Wilson loops 1/6 BPS Wilson loops
1/2 BPS Wilson loops
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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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The End