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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 Presentation
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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
𝑈 (𝑁+𝑙 )𝑘×𝑈 (𝑁 )−𝑘=𝑈 (𝑁 )𝑘×𝑈 (𝑁+𝑘−𝑙 )−𝑘Seiberg duality
dual to Vasiliev higher spin theory?
“ABJ triality” [Chang+Minwalla+Sharma+Yin]
𝐻3 (𝑆7/ℤ𝑘 ,ℤ )=ℤ𝑘
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
𝑁1 𝑁 2
𝐴1 , 𝐴2
𝐵1 ,𝐵2
D3
D3
NS5 5
9
Lagrangian
ABJ MM [Kapustin-Willett-Yaakov]
Partition function of ABJ theory on :vector multiplets
matter multiplets
10
Pert. treatment — easy Rigorous treatment — obstacle:
Strategy“Analytically continue” lens space () MM
𝑍 ABJ (𝑁1 ,𝑁 2 )𝑘∼𝑍 lens (𝑁1 ,−𝑁 2)𝑘Perturbative relation: [Marino][Marino+Putrov]
Simple Gaussian integral Explicitly doable!
11
Expression for
𝑞≡𝑒−𝑔𝑠=𝑒− 2𝜋 𝑖𝑘 ,𝑁≡𝑁1+𝑁2
: -Barnes G function(1−𝑞 )
12 𝑁 (𝑁− 1)
𝐺2 (𝑁+1 ;𝑞)=∏𝑗=1
𝑁−1
(1−𝑞 𝑗 )𝑁− 𝑗
partition of into two sets
12
13
Now we analytically continue: 𝑁 2→−𝑁 2
It’s like knowing discrete data𝑛 !
and guessingΓ (−𝑧)
Analytically cont’d
, Analytic continuation has ambiguity
criterion: reproduce pert. expansion Non-convergent series — need regularization
-Pochhammer symbol:
14
: integral rep (1)
Finite “Agrees” 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 “mirror
expression”
Vanishes for
18
— Starting point for Fermi gas approach
— Agrees with ABJ conjecture
Details / Example
𝑍 lens
𝑞≡𝑒−𝑔𝑠=𝑒− 2𝜋 𝑖𝑘 ,𝑁≡𝑁1+𝑁2
partition of into two sets
20
𝑁1=1
21
𝑁1=1
𝑆 (1 ,𝑁2)
22
How do we continue this ?Obstacles:1. What is
2. Sum range depends on
-Pochhammer & anal. cont.
23
For (𝑎 )𝑛≡ (1−𝑎 ) (1−𝑎𝑞 )⋯ (1−𝑎𝑞𝑛−1 )=
(𝑎 )∞(𝑎𝑞𝑛)∞
.
For ,(𝑎 )𝑧≡
(𝑎 )∞(𝑎𝑞𝑧 )∞
.
In particular,(𝑞 )−𝑛≡
(𝑞 )∞(𝑞1−𝑛 )∞
=(1−𝑞) (1−𝑞2)⋯
(1−𝑞1−𝑛) (1−𝑞2−𝑛)⋯ (1−𝑞0 )⋯=∞
Good for any Can extend sum range
Comments -hypergeometric function
Normalization𝑍 lens (1 ,𝑁2 )𝑘=
1𝑁2 !
∫… 𝑍 lens (1 ,−𝑁 2 )𝑘=1
Γ (1−𝑁2)∫…
Need to continue “ungauged” MM partition functionAlso need -prescription:
24
𝑍 ABJ (1 ,𝑁 2 )𝑘
𝑵𝟐=𝟏:
𝑵𝟐=𝟐∑𝑠=0
∞
(−1 )𝑠 1−𝑞𝑠+1
1+𝑞𝑠+1 =∑𝑠=0
∞
(−1 )𝑠[ 𝑠+12
𝑔𝑠−(𝑠+1 )3
24𝑔𝑠3+⋯ ]
∑𝑠=0
∞
(−1 )𝑠=12
𝑍 ABJ (1,1 )𝑘=1
4∨𝑘∨¿¿
𝑞=𝑒−𝑔𝑠
¿18 𝑔𝑠+
1192 𝑔𝑠
3+⋯
Li𝑛 (𝑧 )=∑𝑘=1
∞ 𝑧𝑘
𝑘𝑛,Li−𝑛 (−1 )=∑
𝑘=1
∞
(−1 )𝑠𝑠𝑛
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
“ ”
But more!
Non-perturbative completion
28
: non-perturbative
poles at
𝑠=𝑘2 − 𝑗
𝑠=𝑘− 𝑗𝑗=1 ,…,𝑁2−1
zeros at
Integrand (anti)periodic
Integral = sum of pole residues in “fund. domain”
Contrib. fromboth P and NP poles
𝑘2
𝑘
𝑁 2−1𝑁 2−1ℜ𝑠
ℑ𝑠
𝑂
𝑠
NPpolesP polesNPzeros
fund. domain
Contour prescription
29
𝒌𝟐−𝑵𝟐+𝟏≥𝟎
𝒌𝟐−𝑵𝟐+𝟏<𝟎
Continuity in fixes prescription Checked against original ABJ MM integral for
𝑘2
𝑘
𝑁 2−1𝑁 2−1
𝑂−𝜖
𝑠
𝑘2
𝑘𝑘2 −𝑁 2+1−𝜖 migrat
e
𝑠
large enough:
smaller:
𝑁1≥2
30
Similar to Guiding principle: reproduce pert. expn.
Multiple -hypergeometric functions appear
Seiberg duality
31
𝑁
𝑙
NS5 5
𝑈 (𝑁+𝑙 )𝑘×𝑈 (𝑁 )−𝑘=𝑈 (𝑁 )𝑘×𝑈 (𝑁+¿ 𝑘∨−𝑙 )−𝑘
𝑁
¿𝑘∨−𝑙
NS5 5
Cf. [Kapustin-Willett-Yaakov]
Passed perturbative test.What about non-perturbative one?
Seiberg duality:
32
𝑈 (1 )𝑘×𝑈 (𝑁 2 )−𝑘=𝑈 (1 )−𝑘×𝑈 (2+|𝑘|−𝑁2 )𝑘Partition function:
duality inv. by“level-rank duality”
Odd
33
(original)
Integrand identical (up to shift), and so is integral
P and NP poles interchanged
52
5𝑂
(dual)
52
5𝑂
52
5𝑂
52
5𝑂
Even
34
(original)
(dual)
2 4𝑂 2 4𝑂
2 4𝑂2 4𝑂
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 “mirror description” 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 Vasiliev’s higher spin
theory Toward membrane dynamics?
Thanks!