日露ミニワークショップ

23/03/2013

Non-Perturbative Effects
in ABJM Theory
Yasuyuki Hatsuda (DESY & TITECH)
Collaborators:
Sanefumi Moriyama (KMI, Nagoya U.)
Kazumi Okuyama (Shinshu U.)
Based onarXiv: 1207.4283, 1211.1251 & 1301.5184

1. IntroductionABJM Theory•

In this talk, I will talk about non-perturbative
corrections to the ABJM partition function•What is ABJM?

[Aharony, Bergman, Jafferis, Maldacena ’08]

3d superconformal Chern-Simons-matter theory
with gauge group U (N )k ⇥ U (N ) k•

There are two independent parametersRank of gauge group N‘t Hooft coupling&= N/k&String coupling gs = 2⇡i/k

Chern-Simons level k3

•Why important?
‣Low energy effective theory on multiple M2 branes‣

has gravity dual (AdS/CFT correspondence)
M-theory on AdS4 ⇥ S 7 /Zk (k ⌧ N 1/5 )
3AdS
⇥
CP
Type IIA on
4‣•(‘t Hooft limit)

Deep mathematical structure (integrability etc.)

Recently, there is much progress in exact computationof some quantities (partition function, VEVs of BPS
3Wilson loops, etc.) in ABJM theory on S[Kapustin, Willett, Yaakov ’09 etc.]

4ABJM Partition Function•By using the localization, the partition function isexactly reduced to finite dimensional matrix integral[Kapustin, Willett, Yaakov ’09]

Infinite dimensional path integral
Localization (exact reduction)
Finite dimensional matrix integral
ZABJM (N ) =1
N !2ZNd µNd ⌫(2⇡)N (2⇡)NQµµ⇥ exp5⌫⌫

ijij
22[2sinh()][2sinh()]i<j22Qµi ⌫ j 2)]
i,j [2 cosh(2"ikNX

4⇡ i=1(µ2i⌫i2 )#•It is still non-trivial to extract analytical properties,
e.g. large N behavior, from this matrix integral•

Traditional matrix model (& topological string)
techniques are useful to study it•

However, they are very technical (and it seems hardfor non-specialists to join the development)•In this talk, we use another simple method tocompute the partition function called Fermi-gas
approach (which does not require any knowledge[Marino, Putrov ’11]
about matrix model)
6Known Results•

In the ‘t Hooft limit, the free energy has the genus
expansionN, k ! 1
F ( , gs ) ==log Z( , gs )

1
Xgs2g

g=0•

2: finite

Fg ( ) + O(e⇡kp2)

At strong ‘t Hooft coupling, the genus g free energy
is divided into two parts
Fg ((p)) = Fg ((np)
Fg ((np)) + Fg (p2⇡ 2) ⇠ O(e)

7)

(1)

•

In the large N limit, the genus zero free energy
3/2[Drukker, Marino, Putrov ’10]
shows N
behavior

F0 ( )
gs2•⇡p⇡ 2k3

NFrom gravity dual
[Klebanov, Tseytlin ’96]

3/2

The “perturbative” part can be summed up in all[Fuji, Hirano, Moriyama ’11]
genus
1Xg=0

gs2g

2Fg(p) () ⇠ log Ai

"✓

⇡2 k2

◆1/3 ✓Nk

1

243k

◆#

•

These results were first derived by using the matrix
model technique•

The same results are simply re-derived from theFermi-gas approach8

Non-Perturbative Corrections•The “perturbative” part is understood•The next step is to consider non-perturbativecorrections
F ( , gs ) =1X

g=0••gs2g

2Fg(p) ( ) + O(e2⇡p2) + O(e⇡kp

There are two kinds of corrections‣Worldsheet instantons: non-perturbative in ↵0 = 1/‣

D2 instantons: non-perturbative in gs = 2⇡i/kp

Our goal is to understand the analytical structure ofsuch non-perturbative corrections
92)

Holographic Picture•

These two kinds of instantons have clear interpretationin the gravity dual
M2 wrapping M

M2 wrapping WM

7S /ZkWCPRP3
D2 wrapping RP3[Drukker, Marino, Putrov ’11]

1CP3F1 wrapping CP1[Cagnazzo, Sorokin, Wulff ’09]
10

Plan
1. Introduction
2. Review
3. Non-Perturbative Corrections
4. Summary

11

2. Review

Mirror of ABJM•

The ABJM theory at k = 1 has a mirror description
ABJM at k = 1•

N = 4 SYM with an adjoint &a fundamental hypermultiplet

The partition function of the “mirror” theory
Zmirror (N ) =•Mirror

1
2N N !

ZNd x(2⇡k)NQj<k tanhQ

2 xj(xk)
2kxi)
2cosh(i2One can directly show the exact equalityZABJM (N ) = Zmirror (N ) [Kapustin, Willett, Yaakov ’10]

•For k 6= 1 , the field theoretical interpretation of the“mirror” partition function is unclear
13

Fermi-gas Approach•

Using Cauchy determinant, the mirror partition
function is written as
Z(N ) =1N!

⇢(q1 , q2 ) =

•

X( 1)

✏( )

2SN1ZNd qNY(2⇡k)N i=1
1⇢(qi , q (i) )1·q1 1/2 ·q1 q2(2 cosh 2 )2 cosh( 2k ) (2 cosh q22 )1/2An advantage of this expression is that the grandpartition function is just the Fredholm determinant[Okuyama ’11; Marino, Putrov ’11]for ⇢⌅(µ) =1XZ(N )eµN

N =014µ= Det(1 + e ⇢)•

Mathematically, our problem is how to compute theFredholm determinantµµ⌅(µ) = Det(1 + e ⇢) = exp [Tr log(1 + e ⇢)]" 1#X ( 1)n 1= expenµ Tr ⇢nnn=1•

The grand partition function is also interpreted as that
[Marino, Putrov ’11]
for an ideal Fermi-gas⌅(µ) =1Y(1 + e(Ejµ))j=0eĤ= ⇢ˆ⇢(x1 , x2 ) = hx1 |⇢|xˆ 2i15

Exact Partition Function•

For some integers k, we can exactly computenTr ⇢ =dq11 2⇡k···dqn2⇡k⇢(q1 , q2 )⇢(q2 , q3 ) · · · ⇢(qn , q1 )[YH, Moriyama, Okuyama ’12;
Putrov, Yamazaki ’12]

At k = 1, for example,
Z1 (1) =Z1 (4) =Z1 (6) =14,

Z1 (2) =116⇡

51512⇡ 21024,,Z1 (3) =Z1 (5) =1312124576⇡ 3147456⇡

3+64⇡132048⇡++2164591024⇡4096

14096···•Z 1

Z1 (44) =11719390154645749
13914455051018076713858888929213810698564403200⇡ 2216+ ···•Exact results indeed show the N200

F =log Z

k=63/2k=4 k=3behavior
k=1k=2

150

100

Exact data
“Perturbative” sum
50

Z (pert) (k, N ) =

✓2

⇡ k2

◆1/3eA(k) Ai

"✓

2

⇡ k2

◆1/3 ✓Nk

1

243k

◆#

N 3/2
0020

4060

80
17100

120140Large N Behavior•To know the large N behavior, we consider the large µbehavior of the grand potential
J (µ) =1X( 1)nn=1•1nenµ Tr ⇢n

Quantum mechanics⇢ˆ =•1~ ⌘ 2⇡k[q̂, p̂] = i~⇢(x1 , x2 ) = hx1 |⇢|x
ˆ 2i
11(2 cosh q̂2 )1/2 2 cosh( p̂2 ) (2 cosh q̂2 )1/2

Classical limit (k ! 0)
Tr ⇢n ⇡

1kZdqdp(2⇡)2

12 cosh( q2 ) · 2 cosh( p2 )18n=1(2⇡)2 k

4 n(2)2 (n)Jcl (µ) ==•eµ42µ33⇡+2µ32µ ◆1 1 13 e, , ; 1, ;2 2 22 16+2⇣(3)+ O(e⇡2233⇡ 2 kµ +✓e2µ428⇡2µSemi-classical result
J (µ) =

•3 F2✓)

F3✓2µ ◆

3 3
e1, 1, 1, 1; , , 2;
2 216(µ ! 1)

[Marino, Putrov ’11]

13k+k24◆µ + A(k) + O(e2µ,e4µ/k

The partition function is given byZ(N ) =Zdµ2⇡ieJ(µ)

Nµ•In the large N limit, the saddle-point approximationis good•

The saddle-point equation:
J 0 (µ⇤ )µ⇤ ⇡ ⇡N =019rkN2)

•

3/2The N
behavior is derived from the saddle-point
approximation
F (N ) ⇡•⇡ 2k3

N

3/2

The “perturbative” result is also reproduced simplyZ•

J (µ⇤ ) + µ⇤ N =p(pert)(N ) =

Z=✓dµ2⇡i

⇡2 k2

exp

◆1/3

e

2
3⇡ 2 k

A(k)

µ✓

3Ai

"✓

N

⇡2 k
21k

3k

24

◆1/3 ✓N◆µ + A(k)k1243k◆#Lesson: The grand canonical ensemble simplifies theproblem!
20

3. Non-PerturbativeCorrections

Non-Perturbative Effects•

Let us proceed to consider the non-perturbative
corrections to the ABJM partition function•The grand canonical ensemble is useful tounderstand these corrections•There are two kinds of corrections
2µ‣

D2 instantons: corrections like O(e‣Worldsheet instantons: corrections likeO(e22) ⇠ O(e4µ/k) ⇠ O(e⇡kp2⇡

2p2))

•

These two corrections have the following generic[Marino, Putrov ’11]
forms
J D2 (k, µ) =JWS(k, µ) =1 hX`=11Xia` (k)µ2 + b` (k)µ + c` (k) edm (k)e2`µ

4mµ
km=1

•

Our goal is to determine these coefficients•

In general, there are also “bound states” of these two(2`+ 4mk )µ )instantons that induce the corrections O(eJ(np)(k, µ) =1X

f`,m (k, µ) exp`,m=0(`,m)6=(0,0)



✓D2 instantons: m = 0
worldsheet instantons: ` = 023

2` +4mk`

M

◆

µ ,mW

Worldsheet Instantons•

The worldsheet instanton corrections can becomputed by using duality
ABJM matrix model
Analytic continuation

Lens space matrix model
Large N duality
11P⇥
P
Topological string on local•

Using this relation,dm (k) =1 X
Xg=0 d|m[YH, Moriyama, Okuyama ’12]m ⇣(1)dgndm

2 sin2⇡m ⌘2g2dk(Weighted sum of) Gopakumar-Vafa invariants24•

We can return to the correction to the partition function⇡ 2 k 1/3
Ai[( 2 ) (N1(1-WS)
Z(N ) =2 2⇡⇡ 2 k 1/3
sin ( k ) Ai[( 2 ) (Nk11+)]243kk1)]243k

Z(N ) = Z (pert) (N )(1 + Z (np) (N ))•ˆ=Let us consider the ‘t Hooft limit
F(1-WS)=2gs e2⇡p2ˆ+✓135p6 2⇡p1
ˆ4⇡ 2 ˆ◆e2⇡p2ˆ

124+ O(gs2 )agree with [Drukker-Marino-Putrov ’10]

•Our result is understood as the all-genus resummationof each instanton
Fg ( ) = Fg(p) ( ) + Fg(1-WS) ( ) + Fg(2-WS) ( ) + · · ·
1X

gs2g

2Fg(n-WS) ( ) = F (n-WS) ( , gs )g=025

Remarks•

The worldsheet instanton corrections diverge atsome levels•

However, the total partition function is finite even atthese levels (e.g. k = 1)

•Why?

•

Clue: At these levels, the worldsheet instantoncorrections are just the same order as the D2instanton corrections•Natural expectation: The D2 instanton corrections
also diverge at these levels, and both poles cancel
each other26

D2 Instantons•

Using this pole cancellation mechanism, we canknow the D2 instanton corrections
2µ•

Let us consider the correction O(e•There are two contributions: 1-D2 instanton and nworldsheet instanton•The n-worldsheet instanton correction is divergent
at k = 2n
lim dn (k)e

k!2n

•4n
k µ= ( 1)n1

4n
⇡ 2 (k2n)+24(µ + 1)
⇡ 2 (k2n)+) at k = 2n2µ2 + 2µ + 1n⇡ 2

These poles should be canceled by the 1-D2instanton correction
27+ wn e2µ•

The pole cancellation gives strong constraints
lim a1 (k) = O(1)k!2nlim b1 (k) = ( 1)4n+ O(1)⇡ 2 (k2n)
4nn 1
lim c1 (k) = ( 1)k!2n
⇡ 2 (k 2n)2k!2n•

4⇡ 2 (k2n)+ O(1)

Moreover, the 1-D2 instanton correction has thefollowing small k expansion[Marino, Putrov ’11]
a1 (k) =b1 (k) =c1 (k) =4⇡2 k+45k

⇡2 k
✓
46

⇡22k⇡ 2 k3

296+

◆

13 k67⇡ 2 k3+ O(k5 )1440+ O(k5 )

⇡2112

k28

⇡ 2 (104 + 5⇡ 2 )2880

k3 + O(k5 )•From these two conditions, we can conjecture theanalytic form of the 1-D2 instanton correction[YH, Moriyama, Okuyama ’12]4a1 (k) =⇡2 kcos⇣ ⇡k ⌘2⇣ ⇡k ⌘⇣ ⇡k ⌘b1 (k) = csccos2⇡22⇣ ⇡k ⌘⇣ ⇡k ⌘⇣ ⇡k ⌘25k1kc1 (k) =++ cot+ csc2cos
3k12
⇡
22222

•

These expressions correctly reproduce the higher
order corrections in the small k expansions[Calvo, Marino ’12]

•The corrections up to 4-instanton was also fixed[Calvo, Marino ’12; YH, Moriyama, Okuyama ’13]
29

Bound States•

The “bound state” contribution is also captured bythe pole cancellation condition
J(np)(k, µ) =1X

f`,m (k, µ) exp`,m=0(`,m)6=(0,0)

4µ✓2` +4mk◆µ ,•O(e•Three contributions: (`, m) = (2, 0), (1, n), (0, 2n)•) correction at k = 2n

unknown

Conjecture
f1,n (k, µ) =known

2n⇡ 2 a1 (k)dn (k)30•General bound statesf`,m =X[YH, Moriyama, Okuyama ’13]

1

p!p!
···p!
12
`(p ,...,p )1dm`2A`,m =p1 + 2p2 + · · · + `p` = `,•p`p1
p2A1,m A2,m · · · A`,mp12m⇡ a` (k)0, p20, . . . , p`

Sum up all the bound states
J (np) (k, µ) = J D2 (k, µ) + J WS (k, µe↵ )1X⇡ k2µe↵ = µ +231`=1a` (k)e2`µ0

4. Summary

Summary•

The grand canonical ensemble of the corresponding
Fermi-gas is useful to study the ABJM partition
function systematically•

We have computed the exact values for some levels•

The non-perturbative corrections can also beanalyzed in the grand canonical ensemble•Pole cancellation conditions strongly constrain theforms of D2 instanton corrections (and bound states)•It is very hard to treat the D2 instantons in the usual[Drukker, Marino, Putrov ’11]
matrix model technique33

Future Direction 1•

Application to BPS Wilson loops[Klemm, Marino, Schiereck, Soroush ’12]
(1/2)
hW⇤
i=

•

12csc

✓

2⇡k◆

⇡ 2 k 1/3
Ai[( 2 ) (N⇡ 2 k 1/3
Ai[( 2 ) (N7p)]
3k⇡ 2N/k⇠O(e)1)]
3kk24
k24

We can compute exact values for some levels[YH, Honda, Moriyama, Okuyama, work in progress](1/2) (k=3)
iN =1 = 4hW⇤

log W4((1/2) (k=3)
hW⇤iN =2 =p10

3 + 2 3)⇡3( 3 + ⇡)4(9 + ((1/2) (k=3)
hW⇤iN =3 =p6

27 + 14 3)⇡)p
9 + (108 64 3)⇡(1/2) (k=3)
hW⇤iN =4 =

p

k=38

p

4

2

16⇡( 9(5 + 2 3) + 14 3⇡)p
378 + 180⇡ + ( 539 + 256 3)⇡ 25-2•

How about non-perturbative corrections?
341015N

Future Direction 2•

Systematic way to compute the D2 instanton[YH, Marino, Moriyama, Okuyama, work in progress]
corrections?

•11P⇥
PConnection with topological string on locala` (k) =

11 X

k n=0

Ja (k, µ) =1X(n) 2na` k ,a` (k)e2`µb` (k) =,

11 X

k n=0

Jb (k, µ) =`=1(n) 2nb` k

1Xb` (k)e2`µ`=1ABJM

Topological string

Leading corrections Ja(0) (µ), Jb(0) (µ)

Classical A- & B-periods

Full corrections Ja (k, µ), Jb (k, µ)“Quantum” A- & B-periods

Chern-Simons level k ⇠ ~Relation between µ & µe↵✏1 ⇠ ~(✏2 = 0)Mirror map35

Thank you!