Exact results on the ABJ theory
Kazumi Okuyama
Shinshu U

JSPS/RFBR workshop
work in progress with Masazumi Honda

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

1 / 15ABJ(M) Theory on S 3
Holographic duality
M-theory on AdS4 × S 7 /Zk ↔ ABJ(M) theory on S 3

ZM-theory = ZABJ(M)This theory also has an interesting relation toNekrasov-Shatashvili limit of refined topological string on local P1 × P1
I quantum curve and quantum period[Hatsuda-Mariño-Moriyama-KO]
IKazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

2 / 15ABJ(M) Matrix Model
Partition function of U(N1 )k × U(N2 )−k ABJ theory
Z = e iθZik22d N1 µd N2 ν e 4π (µ −ν )(2π)N1 +N2 N1 !N2 !

"Qνi −νj #2µi −µj·2sinh
22µi −νji,j 2 cosh 2i<j 2 sinh
QWe consider U(N + M)k × U(N)−k theory with small M, large N limit
Free energy exhibits the N 3/2 behavior [Drukker-Mariño-Putrov]" √#π 2k3/2Z ≈ exp −(N − B)
3

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

3 / 15

Exact Values of Z
We are interested in the instanton corrections to the free energy
To study the instanton corrections,
we have computed the exact partition function at finite N
ABJM: U(N)k × U(N)−k (M = 0)I

[Hatsuda-Moriyama-KO]

up to N = 44, 20 for k = 1, 2, respectively (and few other k’s)

ABJ: U(N + M)k × U(N)−k (M = 1, 2, 3)
I[Honda-KO]

up to N = 62, 57 for (k, M) = (2, 1), (4, 1) (and few other (k, M)’s)

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

4 / 15

Free Energy of ABJM Theory
Free energy exhibits the N 3/2 behavior even at small N
F
200

150

10050

N 3200

Kazumi Okuyama (Shinshu U)

2040

6080100

Exact results on the ABJ theory

120140

JSPS/RFBR workshop

5 / 15

Free Energy of ABJ Theory
Free energy of ABJ theory behaves similarly as the ABJM case
√π 2kF ≈(N − B)3/23F(k,M) =(6,1),(6,2),(6,3)

350(k,M) =(4,1),(4,2)

300(k,M) =(3,1)

250(k,M) =(2,1)

200
150
100
50
20Kazumi Okuyama (Shinshu U)

40

60

80

Exact results on the ABJ theory

100120

N3/2

JSPS/RFBR workshop

6 / 15Grand Canonical Partition Function
Partition function of ABJ theory has the following structure[Awata-Hirano-Sigemori] [Honda]
0b (N, M, k)
Z (N, M, k) = e iθ ZCS (M, k)ZA useful way to analyze ABJ theory is to consider
the grand canonical ensemble [Matsumoto-Moriyama] [Honda-KO]
Ξ(µ) =∞Xb (N, M, k)e Nµ Z

N=0Grand partition function is written as a Fredholm determinant
Ξ(µ) = Det(1 + e µ−H )Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

7 / 15

Hamiltonian of ABJ Fermi Gas
Hamiltonian of ABJ Fermi gas (for even M)
1e −H = e − 2 U(x)1e −U(x) =2 cosh x2

1− 1 U(x)pe 22 cosh 2
M−12Ytanhm=− M−12x + 2πim2k[x, p] = 2πik
Chern-Simons level k= ~ of Fermi gas system

Eigenvalue problem of H is quite interesting
work in progress [Honda-Källén-KO]

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

8 / 15

Relation to Topological String

ABJ(M) theory on S 3
analytic continuation
Pure Chern-Simons theory on S 3 /Z2
geometric transition
Topological string on local P1 × P1

Grand potential J(µ) = log Ξ(µ) is essentially identified withthe free energy of toplogical string on local P1 × P1

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

9 / 15Structure of Grand Potential
J(µ) = log Ξ(µ) has parturbative and non-perturbative parts
J(µ) = Jpert (µ) + Jnp (µ)
Perturbative part is given by an integral of Kähler form
ZZ
1
1
13Jpert (µ) = 2 J +−1J ∧ c26gs
24 gs2
J = T1 E1 + T2 E2 , T1,2 = T ± ib
12M− , gs =T = 2gs µ, b =k2kSeiberg duality M ↔ k − M is realized as the exchange of two P1 ’s

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

10 / 15

Instanton Corrections
There are three types of instanton corrections in Jnp (µ)
123Worldsheet instanton e −T = e −2gs µ⇔ Un-refined (ordinary) topological string Ftop (T1 , T2 , gs )Membrane instanton e −T /gs = e −2µ⇔ NS limit of refined topological string FNS ( Tgs1 , Tgs2 , g1s )⇔ quantum B-period(n, m)-bound state e −(n+m/gs )T⇔ absorbed into worldsheet instanton via quantum mirror map
effT1,2= T1,2 + gs ΠA

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

11 / 15

Quantum Periods
Quantum periods are determined by the wavefunction of D-brane[Mironov-Morozov, Aganagic-Cheng-Dijkgraaf-Krefl-Vafa]
1

H(x, ~∂x )Ψ(x) = µΨ(x), Ψ(x) = e ~ S(x;~,µ)II
ΠA = dS, ΠB =dSAB

Quantum periods are the ~-deformed version of the classical periodsof the mirror curve of local P1 × P1
I

quantum A-period ΠA gives the quantum mirror mapIquantum B-period ΠB is related to the derivative of FNS

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

12 / 15

Non-Perturbative Grand Potential
Combining various analysis (exact Z, small k expansion, numericalfitting, etc.), we arrived at an elegant formulaJnp (µeff ) = Ftop (Teff , gs ) +Ftop (Teff , gs ) = −∞ X
X
m=1 jL ,jR ,dFNS

Teff 1,gs gs
=∞ X
X
n=1 jL ,jR ,dqs = e 2πigs ,1 ∂Teff 1
gs FNS,2πi ∂gsgs gs

NjdL ,jR(2jR + 1)χjL (qsm ) e −md·Teffm/2−m/2 2m(qs − qs)
nd·Teff
χjL (q n/2 )χjR (q n/2 ) e − gsdNjL ,jRn2
q n/2 − q −n/2

q = e 2πi/gs

Instanton coefficients are determined by the BPS invariants NjdL ,jRKazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

13 / 15

HMO Cancellation Mechanism
Worldsheet instanton coefficient diverges at gs = n/m(in particular, for a physical k ∈ Z)
This divergence is precisely canceled by the membrane instanton[Hatsuda-Moriyama-KO]
hilim (worldsheet inst) + (membrane inst) = finite
gs →n/mFinite part is completely fixed by the structure Ftop + ∂FNS
Ftop + ∂FNS works for both ABJM and ABJ

Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory[Honda-KO]

JSPS/RFBR workshop

14 / 15

Conclusions
Membrane instanton are determined by the quantum periods
Effect of bound states is absorbed into the worldsheet instantons bythe quantum mirror map
Membrane instanton corrections to the ABJ(M) grand potential isgiven by the NS limit of refined topological string
Quantum Picard-Fuchs equation for quantum periods?
Higher spin limit? (M, k  N)work in progress [Hirano-Honda-KO-Shigemori]Kazumi Okuyama (Shinshu U)

Exact results on the ABJ theory

JSPS/RFBR workshop

15 / 15