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!