the partition function of abj theory
DESCRIPTION
Masaki Shigemori (KMI, Nagoya U). The Partition Function of ABJ Theory. with Hidetoshi Awata , Shinji Hirano, Keita Nii 1212.2966, 1303.xxxx. Amsterdam, 28 February 2013. Introduction. AdS 4 /CFT 3. Low E physics of M2-branes on Can tell us about M2 dynamics in principle (cf. ) - PowerPoint PPT PresentationTRANSCRIPT
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
3
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
4
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
5
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]
6
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
8
ABJ theory CSM theory
Superconformal IIB picture
π1 π 2
π΄1 , π΄2
π΅1 ,π΅2
D3
D3
NS5 5
9
Lagrangian
ABJ MM [Kapustin-Willett-Yaakov]
Partition function of ABJ theory on :vector multiplets
matter multiplets
10
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!
11
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
12
13
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:
14
: integral rep (1)
Finite βAgreesβ with the
formal series Watson-Sommerfeld
transf in Regge theory
poles from
15
: integral rep (2)
Provides non-perturbative completion
Perturbative (P) poles from
Non-perturbative (NP) poles,
16
Comments (1)Well-defined for all No first-principle derivation;a posteriori justification:
Correctly reproduces pert. expansions
Seiberg duality Explicit checks for small
17
Comments (2) (ABJM) reduces to βmirror
expressionβ
Vanishes for
18
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
20
π1=1
21
π1=1
π (1 ,π2)
22
How do we continue this ?Obstacles:1. What is
2. Sum range depends on
-Pochhammer & anal. cont.
23
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:
24
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
26
Integral rep
Reproduces pert. expn.
27
A way to regularize the sum once and for all:
Poles at , residue
β β
But more!
Non-perturbative completion
28
: 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
29
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
30
Similar to Guiding principle: reproduce pert. expn.
Multiple -hypergeometric functions appear
Seiberg duality
31
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:
32
π (1 )πΓπ (π 2 )βπ=π (1 )βπΓπ (2+|π|βπ2 )πPartition function:
duality inv. byβlevel-rank dualityβ
Odd
33
(original)
Integrand identical (up to shift), and so is integral
P and NP poles interchanged
52
5π
(dual)
52
5π
52
5π
52
5π
Even
34
(original)
(dual)
2 4π 2 4π
2 4π2 4π
Integrand and thus integral are identical P and NP poles interchanged, modulo
ambiguity
Conclusions
36
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 (
37
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!