Instanton Effects in ABJM Theory
Kazumi Okuyama
Shinshu U

seminar@Nagoya[arXiv:1207.4283] Y. Hatsuda, S. Moriyama, KO[arXiv:1211.1251] Y. Hatsuda, S. Moriyama, KO[arXiv:1301.5184] Y. Hatsuda, S. Moriyama, KO[arXiv:1306.1734] Y. Hatsuda, M. Mariño, S. Moriyama, KO[arXiv:1306.4297] Y. Hatsuda, M. Honda, S. Moriyama, KO

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Introduction
ABJM theory describes the dynamics of multiple M2-branes on C4 /ZkABJM = 3d U(N)k × U(N)−k Chern-Simons-matter theory
Holographic dual
ABJM on S 3 ↔ M-theory on AdS4 × S 7 /Zk
ABJM theory on S 3 can be thought of as a nonperturbative definitionof M-theory on AdS4 × S 7 /Zk
The partition function Z (N, k) of ABJM theory on S 3 can beobtained by the localization technique [Kapustin-Willett-Yaakov]
I’t Hooft limit (k, N → ∞, N/k = fixed) ⇒ Type IIA on AdS4 × CP3I

We are interested in the M-theory regime k = finite

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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2 / 34ABJM Matrix Model
Partition function of ABJM theory on S 3"Qν −ν #2µi −µj· 2 sinh i 2 jikd N µd N νi<j 2 sinh 2(µ2 −ν 2 )
4πZ (N, k) =eQµ−ν
2N2ij(2π) (N!)
i,j 2 cosh 2Z
Nxi − xj Yd x Y
11= Ntanh2N2k
2 cosh x2i
2 N!
(2πk)
Zi<ji

ABJM matrix model is simple enough to be tractable
At the same time, it captures the essential part of quantum M-theory
ABJM matrix model = ‘Ising model’ of M-theory

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Parameter RegionN
11d SUGRAIIA string

quantum M-theory

tensionlesskThere is no phase boundaryI

CFT dual of M-theory 6= matrix modelI(k, N) parametrizes a family of 3d CFTs

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Large N Behavior of ABJM Matrix Model
Large N limit of ABJM matrix model reproduces the N 3/2 d.o.f.
of N M2-branes [Drukker-Marino-Putrov]!
√π 2k 3/2
NZ (N, k) ≈ exp −3Perturbative part of partition function is given by the Airy function[Fuji-Hirano-Moriyama]h 1i1Zpert (N, k) = C − 3 e A · Ai C − 3 (N − B)
C=

2,π2k

B=k1+24 3k

A is the contribution of constant maps [Hanada et al]

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Large N Expansion in M-theory
M-theory expansion is quite different from the genus expansion ofstring theoryX
string : F =Fg N 2(1−g ) + O(e −N )g

ABJM : F =X`√

3c` N 2 (1−`) + O(e − N ) = Fpert + Fnp

Expansion parameter in quantum M-theory (RAdS = 1)11
∼ 9 ∼ N 3/2 ,G11
`plKazumi Okuyama (Shinshu U)

TM2 ∼Instanton Effects in ABJM Theory

√
1
∼ N
3
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Non-Perturbative Corrections

We are interested in the non-perturbative part of free energy Fnp
Fnp comes from instanton effects√Worldsheet instanton ∼ e −√ N/kMembrane instanton ∼ e − kN

To study the behavior of instanton corrections, we compute Z (N, k)
exactly at finite N ∼ `−6plfinite N ↔ quantum regime of M-theory

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Exact Values of Z (N, k)
N = 2 at arbitrary k [KO]up to N = 9 at k = 1 [Hatsuda-Moriyama-KO]
up to N = 19 at k = 1 [Putrov-Yamazaki]
Larger N at various k [Hatsuda-Moriyama-KO]
up to N = 44, 20, 18, 16, 14 at k = 1, 2, 3, 4, 6, respectively
For example, Z (N) at k = 1 are
1π−31, Z (2) = 4 , Z (3) = 6 ,222 π2 π2−π + 10−9π 2 + 20π + 26
Z (4) =,Z(5)=,210 π 2
212 π 236π 3 − 121π 2 + 78−75π 3 + 193π 2 + 174π − 126
Z (6) =,Z(7)=,214 32 π 3
216 3π 31053π 4 − 2016π 3 − 4148π 2 + 876
Z (8) =, ···221 32 π 4Z (1) =Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Free Energy of ABJM Theory
Free energy exhibits the N 3/2 behavior even at small N√π 2k 3/2NF ≈3F200

150

10050

N 3200Kazumi Okuyama (Shinshu U)

2040

60

80100

Instanton Effects in ABJM Theory

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HMO Cancellation Mechanism
Free energy should be a smooth function of string coupling gs = k2
small gs (IIA) −→ large gs (M)However, the worldsheet instanton coefficient (∼nat some rational number gs = m1) diverges
sin πmgs2

This divergence is precisely cancelled by the membrane instanton[Hatsuda-Moriyama-KO]
hilimn (worldsheet inst) + (membrane inst) = finite
gs → mMariño coined this mechanism as “HMO cancellation”
HMO cancellation + small k expansion ⇒ find membrane instanton
Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Membrane is still mysterious

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Grand Canonical Partition Function
The measure factor is not the Vandermonde determinant, but theCauchy determinant
A useful way to analyze ABJM matrix model is to consider the grandcanonical ensemble [Marino-Putrov, KO]
Ξ(µ) =∞
Xe Nµ Z (N)N=0Grand partition function is written as a Fredholm determinant
Ξ(µ) = Det(1 + e µ−H ) =∞Y(1 + e µ−En )n=0This is interpreted as a system of free Fermi gas
Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Hamiltonian of ABJM Fermi Gas
One-body Hamiltonian of ABJM Fermi gas
111e −H = pp p
2 cosh x2 2 cosh 2 2 cosh x2[x, p] = 2πik
Chern-Simons level kII= ~ of Fermi gas systemsmall k expansion ↔ semi-classical expansion of Fermi gas
large k expansion ↔ genus expansion in gs = 2/k

The n-th eigenvalue En of the Hamiltonian H behaves as
En2 ∼ C −1 nKazumi Okuyama (Shinshu U)(n  1)

Instanton Effects in ABJM Theory

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Classical Fermi Surface
The constant energy surface becomes a square in the large E limit

2 coshp −1x· 2 cosh
= e −E22⇒|x| + |p| = 2E200

1000-100-200-200-1000100200

Number of states n(E ) below E in the semi-classical limit[Marino-Putrov]√Area(2 2E )22== 2 E 2 = CE 2n(E ) ≈2π~2π · 2πkπ kKazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Perturbative Grand Potential
We consider the large µ expansion of grand potential J(µ)J(µ) = log Ξ(µ) = Tr log(1 + e µ−H )Semi-classical expression of grand potential
Z ∞Z ∞dn(E )n(E )Cµ−EJ(µ) ≈dElog(1 + e)≈dE≈ µ3E−µdE
31+e00N 3/2 d.o.f. of M2-brane is easily reproduced from the saddle point
approximation
ZC µ3
C µ323∗
Z (N) ≈ dµ e −Nµ+ 3 ≈ e −Nµ∗ + 3 = e − 3 C µ∗ (C µ2∗ = N)

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Airy Function from Fermi Gas[Marino-Putrov] studied the semi-classical expansion of grandpotential J(µ) in the small k, large µ limit
J(µ) = Jpert (µ) + Jnp (µ)
CJpert (µ) = µ3 + Bµ + A
3Airy function behavior of perturbative partition function is easily
reproduced from the Fermi gas approach
Zh 1i
1Zpert (N) = dµ e −Nµ+Jpert (µ) = C − 3 e A · Ai C − 3 (N − B)

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

seminar@Nagoya

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Relation to Topological String
M-theory on AdS4 × S 7 /Zkholography
ABJM theory on S 3analytic continuation
Pure Chern-Simons theory on S 3 /Z2
geometric transition
Topological string on (diagonal) local P1 × P1
T1,2 = T ± iπ,Kazumi Okuyama (Shinshu U)

T =4µ= 2gs µ,kInstanton Effects in ABJM Theory

gs =2kseminar@Nagoya

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Instanton Corrections
Two types of instantons (Type IIA picture of AdS4 × CP3 )4µ1Worldsheet instanton = F1 wrapping CP1 (e −T = e − k , µ ∼JWS (µ) =∞X√kN)

4mµdm (k)e − km=12Membrane instanton = D2 wrapping RP3 (eJD2 (µ) =− gTs= e −2µ )∞ hiXa` (k)µ2 + b` (k)µ + c` (k) e −2`µ`=1Worldsheet instantons can be computed from topological string
Small k expansion of membrane instanton coefficients can becomputed from the semi-classical TBA [Calvo-Marino]

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

seminar@Nagoya

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Bound State of Instantons
There are additional bound states of worldsheet m-instanton andmembrane `-instanton(`, m) bound state corresponds toM2-brane wrapping ` times on M and m times on W  4mµJ`,m (µ) ∼ exp − 2` +kM

WS7 /ZkCP3
RPKazumi Okuyama (Shinshu U)

3

CP1

Instanton Effects in ABJM Theory

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Numerical Fitting of Grand Potential
We numerically find the instanton corrections using our exact data ofZ (N)We approximate the nonperturbative part of grand potential by! NXmax
C 3NµJnp (µ) ≈ loge Z (N) −µ + Bµ + A
3
N=0Nmax = 44, 20, 18, 16, 14 for k = 1, 2, 3, 4, 6, respectively

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Instanton Corrections at k = 1, 3, 4, 6
4µ2 + µ + 1/4 −4µ52µ2 + µ/2 + 9/16e+−+2e −8µ + · · ·π22π 2 2
84µ + µ + 1/4 20 −4µ 88 − 16 µ
4 −4µ−µ(k=3)
e− e 3 + ···+Jnp= e 3 − 2e 3 +33π 29924µ + 2µ + 1 −2µ 16 −3µ(k=4)
Jnp= e −µ + −+ ···e+ e
2π 23 24µ + 2µ + 1 20 −2µ 88 − 8 µ4 −2µ− 43 µ(k=6)
3− 2e++e− e 3 + ···Jnp= e33π 299(k=1)
Jnp=Blue part agrees with the prediction of topological string dm (k), but
Red part does not. (d3 (4) = 10/3, d4 (3) = d4 (6) = −8)
Discrepancy is due to the bound states of membrane instanton andworldsheet instanton [HMO, Calvo-Marino]
Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

seminar@Nagoya

21 / 34Membrane Instanton Correction
Only little is known about the membrane instantons
We proposed the analytic form of the membrane 1-instanton
hi1-instJD2(µ) = a1 (k)µ2 + b1 (k)µ + c1 (k) e −2µWe require the following three conditions for a1 (k), b1 (k), c1 (k)1pole structure at k = evenih
4mµlim (WSm-inst )e − k + (D21-inst )e −2µ = finite
k→2mπk2

2

proportional to cos

3matching of small k expansion

Kazumi Okuyama (Shinshu U)(no D2 1-instanton at k = odd)

Instanton Effects in ABJM Theory

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Membrane 1-Instanton Coefficients
From these three conditions, we found the analytic form of themembrane 1-instanton coefficients
4πkcos,2π k2πkπk2cos,b1 (k) = cot
π22
25kk1πkπkc1 (k) = −++cos+ cot
πk23k12 2 sin 2π22
a1 (k) = −This proposal was checked up to O(k 15 ) by [Calvo-Marino]

Membrane 2, 3, 4, 5-instantons were determined in a similar manner[Calvo-Marino, HMO]

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theoryseminar@Nagoya

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Bound States and Effective Chemical Potentialpole cancellation + matching finite pieces at k = 1, .., 6⇓The effect of bound states can be incorporated into the worldsheet
instantons by replacing µ by the effective chemical potential µeff
JWS (µ) + (bound states) = JWS (µeff )∞X−1µeff = µ + Ca` (k)e −2`µ`=1This replacement µ → µeff works also for Wilson loops (open stringsector) [Hatsuda-Honda-Moriyama-KO]Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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General Structure
Total grand potential is simplified by rewriting in µeff
J(µ) =Jpert (µ) + JWS (µ) + (bound states) + JD2 (µ)∞ hi
X=Jpert (µeff ) + JWS (µeff ) +b` (k) + ec` (k) e −2`µeffµeff e`=1eb` (k) is written in terms of a` (k) and b` (k)ec` (k) is determined by eb` (k)∂ec` (k) = −k 2∂kKazumi Okuyama (Shinshu U)eb` (k)
2`k!

∂=∂gs

Instanton Effects in ABJM Theoryeb` (k)gs
2`!

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Example of a` (k) and eb` (k) πk 4cos
π2k2
2a2 (k) = − 2 4 + 5 cos(πk)π k
 πk 

8a3 (k) = − 2 cos
19 + 28 cos(πk) + 3 cos(2πk)
3π k2
 πk   πk 2eb1 (k) = cotcos
π221eb2 (k) = cot(πk) 4 + 5 cos(πk)π 3πk   πk 
4eb3 (k) =cotcos
13 + 19 cos(πk) + 9 cos(2πk)
3π22a1 (k) = −gs -dependence:

Kazumi Okuyama (Shinshu U)πkπ=2gs
Instanton Effects in ABJM Theory

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Spectral Curve of ABJM Matrix Model
We will see thata` (k) = quantum A-period,b` (k) = quantum B-period

Spectral curve of ABJM matrix model is a complexified version of theFermi surface H(x, p) = µRe p
200

2EA100B0{2E-100{2E2ERe q-200-200-100

Kazumi Okuyama (Shinshu U)

0100200

Instanton Effects in ABJM Theory

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Quantum Curve
Quantum curve of local P1 × P1 = Fermi surface H(x, p) = µp x · 2 cosh
= e µ,[x, p] = 2πik2 cosh22
By a change of variable (x, p) → (u, v ), this curve is rewritten aseH(u,v ) = e u + e v + z1 e −u + z2 e −v − 1 = 0
2πi[v , u] = ~ = iπk =gs
The moduli z1 , z2 of the curve areTz1,2 = eKazumi Okuyama (Shinshu U)− g1,2s

1= e −2µ q ± 2 ,Instanton Effects in ABJM Theory

2πiq = e ~ = e gs

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Quantum Period
Quantum periods are determined by the wavefunction of probeD-brane [Aganagic-Cheng-Dijkgraaf-Krefl-Vafa]
1eH(u, ~∂u )Ψ(u) = 0, Ψ(u) = exp S(u, ~)~Quantum periods provide us with a systematic way of computing
membrane instanton coefficients a` (k), b` (k)I
ΠA =dS = C−1AI
ΠB =B∞Xa` (k)e −2`µ`=1dS = C −1∞Xb` (k)e −2`µ`=1This correctly reproduces a` (k), b` (k) computed from thesemi-classical TBA
Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Refined Topological String
Refined topological string on local Calabi-Yau manifold X is atwo-parameter extension of the ordinary topological string
Zref (1 , 2 ) = ZM-theory (X × S 1 × C2 )(y , z1 , z2 ) ∼ (y + 2πR, e 2πi1 z1 , e 2πi2 z2 )Unrefined and Nekrasov-Shatashvili limit of free energy are
Ftop (1 ) = Fref (2 = −1 )FNS (1 ) = 2πi lim 2 Fref (1 , 2 )
2 →0We found worldsheet and membrane instantons are given by
WS = Ftop (1 = gs ),Kazumi Okuyama (Shinshu U)

1
D2 ∼ FNS 1 =gs

Instanton Effects in ABJM Theory

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Membrane Instantons and FNS
Quantum A-period defines the generalized flat coordinate via thequantum mirror mapµeff = µ + ΠA
Quantum B-period is related to the derivative of FNS [Aganagic et al]ΠB − Π2A = C −1
If we treat gs and Teff =JD2 (µeff ) is written as∞ hX`=1∞X`=1~ ∂FNSeb` (k)e −2`µeff =2 ∂µeff

4µeff
as independent variables, thenkiTeff 1
1 ∂−2`µeff
egs FNS,µeff b` (k) + ec` (k) e=2πi ∂gs
gs gs

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

seminar@Nagoya

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Non-perturbative ABJM Grand Potential
Non-perturbative part of grand potential is completely determined bythe refined topological string on the diagonal local P1 × P1

Teff 11 ∂
Jnp (µeff ) = Ftop (Teff , gs ) +gs FNS,2πi ∂gsgs gs∞ X
X(2jR + 1)χjL (qsm ) e −md·Teff
Ftop (Teff , gs ) = −NjdL ,jR m/2−m/2 2m(qs − qs)
m=1 jL ,jR ,dFNS

Teff 1,
gs gs


=∞ X
X
n=1 jL ,jR ,dqs = e 2πigs ,nd·Teff
χjL (q n/2 )χjR (q n/2 ) e − gsdNjL ,jRn2
q n/2 − q −n/22πiq = e gsn
This form guarantees the pole cancellation at gs = mKazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Non-perturbative Completion of Topological String

The above form of ABJM grand potential suggests a generalization toarbitrary local Calabi-Yau manifolds
For the general local Calabi-Yau, we have to turn on a discrete B-field
B = iπK for the pole cancellation to work
Teff 11 ∂Fnp = Ftop (Teff +iπK , gs ) +gs FNS,2πi ∂gsgs gsThis is slightly different from the proposal by Lockhart and Vafa

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

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Summary

The membrane instanton coefficients a` (k) and b` (k) are determinedby the quantum A- and B-period, respectively.
Effect of bound states is incorporated into the worldsheet instantons
by the quantum mirror map µ → µeff
Membrane instanton corrections to the ABJM grand potential is givenby the refined topological string in the Nekrasov-Shatashvili limit
We proposed a non-perturbative topological string on arbitrary local
Calabi-Yau manifolds

Kazumi Okuyama (Shinshu U)

Instanton Effects in ABJM Theory

seminar@Nagoya

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