Non-perturbative Effects in ABJM Theory from Fermi Gas Approach

Non-perturbative Effects in ABJM Theory from Fermi Gas Approach Sanefumi Moriyama (Nagoya/KMI) [arxiv:1106.4631] with H.Fuji and S.Hirano [arxiv:1207.4283, 1211.1251, 1301.5184] with Y.Hatsuda and K.Okuyama

Previously in Fuji-Hirano-M Perturbative Terms of ABJM Matrix Model N 1 =N 2 =N Z(N) = in 't Hooft Expansion Sum Up To

Previously in Fuji-Hirano-M Z(N) = Airy Function (Up To Constant Maps & Instanton Effects) cf: [Marino-Putrov, Honda et al] Renormalization of 't Hooft coupling

Previous Method (Analytic Continuation N 2 - N 2 ) Chern-Simons Theory on Lens Space S 3 /Z 2 (String Completion) Open Top A-model on T*(S 3 /Z 2 ) (Large N Duality) Closed Top A on Hirzebruch Surface F 0 = P 1 x P 1 (Mirror Symmetry) Closed Top B on Spectral Curve u v = H( x, y ) Holomorphic Anomaly Equation!!

Today: Non-Perturbative Effects Not Just Worldsheet Instanton Or Membrane Instanton, But Also Their Bound States Not Qualitative Arguments, But Quantitative Analysis

Contents 1. Motivation 2. Fermi Gas 3. Exact Results 4. NonPerturbative Effects 5. Further Direction

Motivation1 M2-brane ABJ(M) N=6 Chern-Simons Theory (N 1,N 2,k) (N 1 +N 2 )/2 M2 with (N 1 -N 2 ) Fractional M2 on C 4 /Z k Special Case No Fractional Branes: N 1 =N 2 =N Flat Space: k=1 IIA (C 4 /Z k CP 3 x R x S 1 ): k= with N/k Fixed

Motivation1 M2-brane Partition Function of M2 WorldVolume Theory N x M2 Z = Airy(N) Exp[-N 3/2 ] DOF N 3/2 Reproduced [Drukker-Marino-Putrov] Also Non-Perturbative Corrections

Non-Perturbative Corrections [Drukker-Marino-Putrov] 't Hooft Expansion in Matrix Model Exp[-2π 2N/k] Identified as String Wrapping CP 1 in CP 3 Borel Sum-like Analysis Exp[-π 2Nk] Identified as D2-brane Wrapping RP 3 in CP 3

Motivation2 From Gaussian To ABJM ABJM CS Super CS q-deform Gauss Superalg

Message from Airy1 Hidden Structure? String Theory (Dual Resonance Model) Veneziano Amplitude String Conformal Symmetry [Virasoro, Nambu] Membrane Theory Free Energy as Airy Function Hidden Structure for Membrane?

Message from Airy2 Trinity? M- Theory Wave Function of The Universe [Ooguri-Verlinde-Vafa] Membrane WorldVolume Theory [Aharony-Bergman-Jafferis-Maldacena] Airy Function Chern-Simons Theory All Genus Partition Function [Fuji-Hirano-M]

Message from Airy3 Ai(N) = (2πi) -1 dμ Exp[μ 3 /3 - μn] Chern-Simons Partition Function? Z CS (N) = DA Exp A da- A A A Grand Potential in Statistical Mechanics? e J(μ) = 1 + N=1 Z(N) e -μn Z(N) = (2πi) -1 dμ Exp[J(μ) - μn] What is the Statistical Mechanical System?

Contents 1. Motivation 2. Fermi Gas 3. Exact Results 4. NonPerturbative Effects 5. Further Directions

ABJM Matrix Model Z(N) = N 1 =N 2 =N

ABJM Matrix Model Z(N) = N 1 =N 2 =N [ sinh sinh / cosh ] 2

Hidden Structure as Fermi Gas 1. Cauchy Determinant sinh sinh / cosh = Det cosh -1 = σ (-1) σ i [2 cosh (μ i -ν σ(i) )/2] -1 2. Trivialization of One Permutation σ,τ (-1) σ+τ i [2 cosh(μ i -ν σ(i) )/2] -1 [2 cosh(μ i -ν τ(i) )/2] -1 = N! σ (-1) σ i [2 cosh(μ i -ν i )/2] -1 [2 cosh(μ i -ν σ(i) )/2] -1 3. Fourier Transform (μ,ν) (p,q) 4. (μ,ν) Gaussian Integration

Hidden Structure as Fermi Gas After Some Calculation,... Non-Interacting Fermi Gas [Marino-Putrov] Z(N) = (N!) -1 σ (-1) σ i dq i q i ρ q σ(i) Density Matrix ρ = e -H ρ = [2 cosh q/2] -1/2 [2 cosh p/2] -1 [2 cosh q/2] -1/2 Statistical Mechanics Approach N 3/2 & Airy Easily(!) Reproduced

Statistical Mechanics Canonical Ensemble: Sum in Conjugacy Class Grand Canonical Ensemble Grand Potential Inversely e J (μ) = 1 + N=1 Z(N) e -μn Z(N) = (2πi) -1 dμ e J (μ) - μn

WKB Analysis ħ(=2πk)-perturbation Systematic Expansion in e -2μ ~ Exp[-π 2Nk] J(μ) = J pert (μ) + J np (μ) J pert (μ) = C k μ 3 /3 + B k μ + A k J np (μ) = l=1 ( α (l) μ 2 + β (l) μ + γ (l) ) e -2lμ (At Most) Quadratic Prefactor (cf. Linearity in "log[2coshq/2] ~ q") (α (l), β (l), γ (l) ) Determined in ħ-perturbation

Contents 1. Motivation 2. Fermi Gas 3. Exact Results 4. NonPerturbative Effects 5. Further Direction

World Records of Exact Values [Hatsuda-M-Okuyama 2012/07] N max = 9 for k=1 [Putrov-Yamazaki 2012/07] N max = 19 for k=1 [Hatsuda-M-Okuyama 2012/11] N max = 44 for k=1 N max = 20 for k=2 N max = 18 for k=3 N max = 16 for k=4 N max = 14 for k=6

Sample (For k=1) Z(1) = 1/4 Z(2) = 1/16π Z(3) = (π-3)/2 6 π Z(4) = (-π 2 +10)/2 10 π 2 Z(5) = (-9π 2 +20π+26)/2 12 π 2 Z(6) = (36π 3-121π 2 +78)/2 14 3 2 π 3 Z(7) = (-75π 3 +193π 2 +174π-126)/2 16 3π 3 Z(8) = (1053π 4-2016π 3-4148π 2 +876)/2 21 3 2 π 4 Z(9) = (5517π 4-13480π 3-15348π 2 +8880π+4140)/2 23 3 2 π 4

Method: Factorization [Tracy-Widom] ρ(q 1,q 2 ) = E(q 1 ) E(q 2 ) [M (q 1 ) + M(q 2 )] -1 ρ n (q 1,q 2 ) = Σ m (-1) m [ρ m E](q 1 ) [ρ n-1-m E](q 2 ) x [M(q 1 ) - (-1) n M(q 2 )] -1

Method: Hankel Matrix Density Matrix ρ: Isospectral to Hankel Matrix ρ 0 0 ρ 1 0 ρ 2 0 0 ρ 1 0 ρ 2 0 ρ 3 ρ 1 0 ρ 2 0 ρ 3 0 0 ρ 2 0 ρ 3 0 ρ 4 ρ 2 0 ρ 3 0 ρ 4 0 0 ρ 3 0 ρ 4 0 ρ 5 ρ = ρ + 0 0 ρ - A Magic Formula For Hankel Matrix [Hatsuda-M-Okuyama] det (1+zρ - ) / det (1-zρ + ) = [(1-zρ + ) -1 E + ] 0 / [E + ] 0

Contents 1. Motivation 2. Fermi Gas 3. Exact Results 4. NonPerturbative Effects 5. Further Direction

Fitting Grand Potential J(μ) = log[ 1 + Nmax N=1 Z(N) e μn ] Strategy: Plot & Fit J(μ) vs J pert (μ) (J(μ)-J pert (μ))/e -4μ/k vs α 1 μ 2 +β 1 μ+γ 1 (J(μ)-J pert (μ)-j np(1) (μ))/e -8μ/k vs α 2 μ 2 +β 2 μ+γ 2 (J(μ)-J pert (μ)-j np(1) (μ)-j np(2) (μ))/e -12μ/k vs α 3 μ 2 +β 3 μ+γ 3

Example: k=1 J(μ) (J(μ)-J pert (μ))/e -4μ (J(μ)-J pert (μ)-j np(1) (μ))/e -8μ (J(μ)-J pert (μ)-j np(1) (μ)-j np(2) (μ))/e -12μ

Oscillatory Behavior!! Grand Potential Original Definition e J(μ) = 1 + N=1 Z(N) e -μn Periodic in μ = μ + 2πi No More in J pert (μ) etc.

To Remedy the 2πi-Periodicity 2πi-Periodic Grand Potential Exp[J(μ)] = N=- Exp[J naive (μ+2πin)] J naive (μ) = J pert (μ) + J np (μ) J(μ) = J naive (μ) + J osc (μ) Results: J osc (μ) = 2 Cos[C k μ 2 + B k - 8/3k] e -8μ/k +... Hereafter, J osc (μ) Abbreviated

Results from Fitting J k=1 (μ) = [(4μ 2 +μ+1/4)/π 2 ]e -4μ + [-(52μ 2 +μ/2+9/16)/(2π 2 )+2]e -8μ + [(736μ 2-152μ/3+77/18)/(3π 2 )-32]e -12μ +... J k=2 (μ) = [(4μ 2 +2μ+1)/π 2 ]e -2μ + [-(52μ 2 +μ+9/4)/(2π 2 )+2]e -4μ + [(736μ 2-304μ/3+154/9)/(3π 2 )-32]e -6μ +... J k=3 (μ) = [4/3]e -4μ/3 + [-2]e -8μ/3 + [(4μ 2 +μ+1/4)/(3π 2 )+20/9]e -4μ +... J k=4 (μ) = [1]e -μ + [-(4μ 2 +2μ+1)/(2π 2 )]e -2μ + [16/3]e -3μ +... J k=6 (μ) = [4/3]e -2μ/3 + [-2]e -4μ/3 + [(4μ 2 +2μ+1)/(3π 2 )+20/9]e -2μ +... up to 7-instanton

Schematically J k=1 (μ) = [#μ 2 +#μ+#]e -4μ + [#μ 2 +#μ+#]e -8μ + [#μ 2 +#μ+#]e -12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e -2μ + [#μ 2 +#μ+#]e -4μ + [#μ 2 +#μ+#]e -6μ +... J k=3 (μ) = [#]e -4μ/3 + [#]e -8μ/3 + [#μ 2 +#μ+#]e -4μ +... J k=4 (μ) = [#]e -μ + [#μ 2 +#μ+#]e -2μ + [#]e -3μ +...... J k=6 (μ) = [#]e -2μ/3 + [#]e -4μ/3 + [#μ 2 +#μ+#]e -2μ +... WS(1) WS(2) WS(3)

Worldsheet Instanton From Top String Implication from Topological Strings J WS k (μ) = m=1 d (m) k e -4mμ/k Multi-Covering Structure d (m) k = g n m (-1) m/n N g n/n (2 Sin[2πn/k]) 2g-2 Gopakumar-Vafa Invariant on F 0 =P 1 xp 1 N g n

Match with Topological String? J k=1 (μ) = [#μ 2 +#μ+#]e -4μ + [#μ 2 +#μ+#]e -8μ + [#μ 2 +#μ+#]e -12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e -2μ + [#μ 2 +#μ+#]e -4μ + [#μ 2 +#μ+#]e -6μ +... J k=3 (μ) = [#]e -4μ/3 + [#]e -8μ/3 + [#μ 2 +#μ+#]e -4μ +... J k=4 (μ) = [#]e -μ + [#μ 2 +#μ+#]e -2μ + [#]e -3μ +...... J k=6 (μ) = [#]e -2μ/3 + [#]e -4μ/3 + [#μ 2 +#μ+#]e -2μ +... WS(1) WS(2) WS(3)

Match with Topological String? J k=1 (μ) = [#μ 2 +#μ+#]e -4μ + [#μ 2 +#μ+#]e -8μ + [#μ 2 +#μ+#]e -12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e -2μ + [#μ 2 +#μ+#]e -4μ + [#μ 2 +#μ+#]e -6μ +... J k=3 (μ) = [#]e -4μ/3 + [#]e -8μ/3 + [#μ 2 +#μ+#]e -4μ +... J k=4 (μ) = [#]e -μ + [#μ 2 +#μ+#]e -2μ + [#]e -3μ +...... J k=6 (μ) = [#]e -2μ/3 + [#]e -4μ/3 + [#μ 2 +#μ+#]e -2μ +... WS(1) WS(2) WS(3) : Match : Divergent : Not-Match

First Guess Membrane Instanton & Worldsheet Instanton, Same Origin in M-theory d k (m) e -4mμ/k = (divergence) + [#μ 2 +#μ+#] e -2lμ Around k = 2m/l Correctly Speaking,...

Cancellation of Divergences? J k=1 (μ) = [#μ 2 +#μ+#]e -4μ + [#μ 2 +#μ+#]e -8μ + [#μ 2 +#μ+#]e -12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e -2μ + [#μ 2 +#μ+#]e -4μ + [#μ 2 +#μ+#]e -6μ +... J k=3 (μ) = [#]e -4μ/3 + [#]e -8μ/3 + [#μ 2 +#μ+#]e -4μ +... J k=4 (μ) = [#]e -μ + [#μ 2 +#μ+#]e -2μ + [#]e -3μ +...... MB(2) J k=6 (μ) = [#]e -2μ/3 + [#]e -4μ/3 + [#μ 2 +#μ+#]e -2μ +... WS(1) WS(2) WS(3) MB(1)

Cancellation of Divergence k=1 k=2 k=3 k=4 k=5 k=6 WS1 WS2 WS3 MB4 MB3 MB2 Worldsheet m-instanton [Sin 2πm/k] -1 Membrane l-instanton [Sin πlk/2] -1 WS4 MB1

More Dynamical Figure k=1 k=2 k=3 k=4 k=5 k=6 WS1 WS2 WS3 MB4 MB3 MB2 Worldsheet m-instanton [Sin 2πm/k] -1 Membrane l-instanton [Sin πlk/2] -1 WS4 MB1

1-Membrane Instanton Vanishing in k=odd Canceling Divergence Matching the WKB data a k (1) = -4(π 2 k) -1 Cos[πk/2] b k (1) = 2π -1 Cot[πk/2] Cos[πk/2] c k (1) =...

How About? (l, m) Bound State? e- l x 2μ - m x 4μ/k Ex: e -3μ Effects in k=4 Sector From Both (0,3) & (1,1) But No Information on Bound States Yet.

2-Membrane Instanton Cancellation of Divergence in (2,0)+(0,2m+1) m k=3 [Calvo-Marino] k=1 a k (2) =... b k (2) =... c k (2) =... l

Bound States (1,m)? Cancellation of Divergence in (2,0)+(1,m)+(0,2m) k=4 m k=2 l

(1,m) Bound States Cancellation of Divergence in (2,0)+(1,m)+(0,2m) Match with (1,1)+(0,3) in k=4 Sector,... (1,m) Bound States J (1,m) k (μ) =... = a (1) k d (m) k e -2μ-4mμ/k

More Bound States Similarly, (2,m) Bound States J (2,m) k (μ) = (a (2) k +(a (1) k ) 2 /2) d (m) k e -4μ-4mμ/k Match with (2,2)+(0,5) in k=3,... (3,m) Bound States J (3,m) k (μ) = (a (3) k +a (2) k a (1) k +(a (1) k ) 3 /6) d (m) k e -6μ-4mμ/k

All Bound States Finally J (l,m) k (μ) = (a (l 1) k ) n 1 /(n1 )!... (a (l L) k ) n L /(n L )! x d (m) k e -2lμ-4mμ/k Sum Over n 1 l 1 +... + n L l L = l (a) n /(n)! Exp[a]??

To Summarize Originally J(μ) = J pert (μ) + J MB (μ) + J WS (μ) + J bnd (μ) J pert (μ) = #μ 3 + #μ + # J MB (μ) = l>0 J MB(l) (μ) = l>0 (#μ 2 + #μ + #) e -2lμ J WS (μ) = m>0 J WS(m) (μ) = m>0 # e -4mμ/k J bnd (μ) = l>0,m>0 J (l,m) (μ)

Effective Chemical Potential μ eff = μ + # l a (l) k e -2lμ Grand Potential J(μ) = J pert (μ eff ) + J' MB (μ eff ) + J WS (μ eff ) J' MB (μ eff ) = l>0 (#μ eff + #) e -2lμ eff Bound States in Effective WS Instanton Membrane Instanton in Linear Functions

Contents 1. Motivation 2. Fermi Gas 3. Exact Results 4. NonPerturbative Effects 5. Further Direction

Further Direction [Work in Progress] Wilson Loops? ABJ Extensions? Understand Generality of Cancellation Understand Membrane Instantons from Matrix Model Terminology Thank You For Your Attention.