The Partition Function of ABJ Theory
description
Transcript of The Partition Function of ABJ Theory
The Partition Function of ABJ
TheoryAmsterdam, 28 February
2013
Masaki Shigemori(KMI, Nagoya U)
with Hidetoshi Awata, Shinji Hirano, Keita Nii1212.2966, 1303.xxxx
Introduction
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AdS4/CFT3
Chern-Simons-matter SCFT
(ABJM theory)
M-theory on
Type IIA on
=
,
Low E physics of M2-branes on Can tell us about M2 dynamics in principle (cf. ) More non-trivial example of AdS/CFT
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ABJ theory Generalization of ABJM [Hosomichi-Lee3-Park, ABJ] CS-matter SCFT Low E dynamics of M2 and fractional M2 Dual to M on with 3-form
π (π+π )πΓπ (π )βπ=π (π )πΓπ (π+πβπ )βπSeiberg duality
dual to Vasiliev higher spin theory?
βABJ trialityβ [Chang+Minwalla+Sharma+Yin]
π»3 (π7/β€π ,β€ )=β€π
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Localization & ABJM Powerful technique in susy field theory
Reduces field theory path integral to matrix model Tool for precision holography ( corrections)
Application to ABJM [Kapustin+Willett+Yaakov] Large : reproduces
[Drukker+Marino+Putrov]
All pert. corrections [Fuji+Hirano+Moriyama][Marino+Putrov]
Wilson loops [Klemm+Marino+Schiereck+Soroush]
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Rewrite ABJ MMin more tractable form
Fun to see how it works Interesting physics
emerges!
Goal: develop framework to study ABJ
ABJ: not as well-studied as ABJM;MM harder to handle
Outline &main results
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ABJ theory CSM theory
Superconformal IIB picture
π1 π 2
π΄1 , π΄2
π΅1 ,π΅2
D3
D3
NS5 5
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Lagrangian
ABJ MM [Kapustin-Willett-Yaakov]
Partition function of ABJ theory on :vector multiplets
matter multiplets
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Pert. treatment β easy Rigorous treatment β obstacle:
StrategyβAnalytically continueβ lens space () MM
π ABJ (π1 ,π 2 )πβΌπ lens (π1 ,βπ 2)πPerturbative relation: [Marino][Marino+Putrov]
Simple Gaussian integral Explicitly doable!
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Expression for
πβ‘πβππ =πβ 2π ππ ,πβ‘π1+π2
: -Barnes G function(1βπ )
12 π (πβ 1)
πΊ2 (π+1 ;π)=βπ=1
πβ1
(1βπ π )πβ π
partition of into two sets
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Now we analytically continue: π 2ββπ 2
Itβs like knowing discrete dataπ !
and guessingΞ (βπ§)
Analytically contβd
, Analytic continuation has ambiguity
criterion: reproduce pert. expansion Non-convergent series β need regularization
-Pochhammer symbol:
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: integral rep (1)
Finite βAgreesβ with the
formal series Watson-Sommerfeld
transf in Regge theory
poles from
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: integral rep (2)
Provides non-perturbative completion
Perturbative (P) poles from
Non-perturbative (NP) poles,
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Comments (1)Well-defined for all No first-principle derivation;a posteriori justification:
Correctly reproduces pert. expansions
Seiberg duality Explicit checks for small
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Comments (2) (ABJM) reduces to βmirror
expressionβ
Vanishes for
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β Starting point for Fermi gas approach
β Agrees with ABJ conjecture
Details / Example
π lens
πβ‘πβππ =πβ 2π ππ ,πβ‘π1+π2
partition of into two sets
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π1=1
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π1=1
π (1 ,π2)
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How do we continue this ?Obstacles:1. What is
2. Sum range depends on
-Pochhammer & anal. cont.
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For (π )πβ‘ (1βπ ) (1βππ )β― (1βπππβ1 )=
(π )β(πππ)β
.
For ,(π )π§β‘
(π )β(πππ§ )β
.
In particular,(π )βπβ‘
(π )β(π1βπ )β
=(1βπ) (1βπ2)β―
(1βπ1βπ) (1βπ2βπ)β― (1βπ0 )β―=β
Good for any Can extend sum range
Comments -hypergeometric function
Normalizationπ lens (1 ,π2 )π=
1π2 !
β«β¦ π lens (1 ,βπ 2 )π=1
Ξ (1βπ2)β«β¦
Need to continue βungaugedβ MM partition functionAlso need -prescription:
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π ABJ (1 ,π 2 )π
π΅π=π:
π΅π=πβπ =0
β
(β1 )π 1βππ +1
1+ππ +1 =βπ =0
β
(β1 )π [ π +12
ππ β(π +1 )3
24ππ 3+β― ]
βπ =0
β
(β1 )π =12
π ABJ (1,1 )π=1
4β¨πβ¨ΒΏΒΏ
π=πβππ
ΒΏ18 ππ +
1192 ππ
3+β―
Liπ (π§ )=βπ=1
β π§π
ππ,Liβπ (β1 )=β
π=1
β
(β1 )π π π
Reproduces pert. expn.of ABJ MM25
Perturbative expansion
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Integral rep
Reproduces pert. expn.
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A way to regularize the sum once and for all:
Poles at , residue
β β
But more!
Non-perturbative completion
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: non-perturbative
poles at
π =π2 β π
π =πβ ππ=1 ,β¦,π2β1
zeros at
Integrand (anti)periodic
Integral = sum of pole residues in βfund. domainβ
Contrib. fromboth P and NP poles
π2
π
π 2β1π 2β1βπ
βπ
π
π
NPpolesP polesNPzeros
fund. domain
Contour prescription
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ππβπ΅π+πβ₯π
ππβπ΅π+π<π
Continuity in fixes prescription Checked against original ABJ MM integral for
π2
π
π 2β1π 2β1
πβπ
π
π2
ππ2 βπ 2+1βπ migrat
e
π
large enough:
smaller:
π1β₯2
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Similar to Guiding principle: reproduce pert. expn.
Multiple -hypergeometric functions appear
Seiberg duality
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π
π
NS5 5
π (π+π )πΓπ (π )βπ=π (π )πΓπ (π+ΒΏ πβ¨βπ )βπ
π
ΒΏπβ¨βπ
NS5 5
Cf. [Kapustin-Willett-Yaakov]
Passed perturbative test.What about non-perturbative one?
Seiberg duality:
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π (1 )πΓπ (π 2 )βπ=π (1 )βπΓπ (2+|π|βπ2 )πPartition function:
duality inv. byβlevel-rank dualityβ
Odd
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(original)
Integrand identical (up to shift), and so is integral
P and NP poles interchanged
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5π
(dual)
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5π
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5π
52
5π
Even
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(original)
(dual)
2 4π 2 4π
2 4π2 4π
Integrand and thus integral are identical P and NP poles interchanged, modulo
ambiguity
Conclusions
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Conclusions Exact expression for lens space MM Analytic continuation: Got expression i.t.o. -dim integral,
generalizing βmirror descriptionβ for ABJM
Perturbative expansion agrees Passed non-perturbative check:
Seiberg duality (
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Future directions Make every step well-defined;
understand -hypergeom More general theory (necklace quiver) Wilson line [Awata+Hirano+Nii+MS] Fermi gas [Nagoya+Geneva] Relation to Vasilievβs higher spin
theory Toward membrane dynamics?
Thanks!