Non-perturbative Effects in ABJM Theory from Fermi Gas Approach

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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

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Previously in Fuji-Hirano-M

Perturbative Terms of ABJM Matrix Model

in 't Hooft Expansion Sum Up To ・・・

Z(N) =

N₁=N₂=N

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Previously in Fuji-Hirano-M

(Up To Constant Maps & Instanton Effects) cf: [Marino-Putrov, Honda et al]

• Airy Function

• Renormalization of 't Hooft coupling

Z(N) =

(4)

Previous Method

(Analytic Continuation N₂ → - N₂)

• Chern-Simons Theory on Lens Space S³/Z₂ (String Completion)

• Open Top A-model on T*(S³/Z₂) (Large N Duality)

• Closed Top A on Hirzebruch Surface F₀ = P¹ x P¹ (Mirror Symmetry)

• Closed Top B on Spectral Curve u v = H( x , y ) Holomorphic Anomaly Equation!!

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Today: Non-Perturbative Effects

• Not Just Worldsheet Instanton Or Membrane Instanton, But Also Their Bound States

• Not Qualitative Arguments, But Quantitative Analysis

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Motivation① M2-brane

Special Case

No Fractional Branes: N₁=N₂=N Flat Space: k=1

IIA (C⁴/Z_k⇒ CP³ x R x S¹): k=∞ with N/k Fixed N=6 Chern-Simons Theory (N₁,N₂,k)

⇕

(N₁+N₂)/2 M2 with (N₁-N₂) Fractional M2 on C⁴/Z_k ABJ(M)

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Motivation① M2-brane

Partition Function of M2 WorldVolume Theory

Z = Airy(N) ≈ Exp[-N^3/2] DOF N^3/2 Reproduced

[Drukker-Marino-Putrov]

N x M2

Also Non-Perturbative Corrections

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Non-Perturbative Corrections

• 't Hooft Expansion in Matrix Model Exp[-2π√2N/k]

Identified as String Wrapping CP¹ in CP³

• Borel Sum-like Analysis

Exp[-π√2Nk]

Identified as D2-brane Wrapping RP³ in CP³

[Drukker-Marino-Putrov]

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Motivation② From Gaussian To ABJM

ABJM

CS Super

Gauss

CS q-deform Superalg

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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?

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Message from Airy② Trinity?

Chern-Simons Theory

M-

Theory

Airy Function

Wave Function of The Universe [Ooguri-Verlinde-Vafa]

Membrane WorldVolume Theory [Aharony-Bergman-Jafferis-Maldacena]

All Genus Partition Function [Fuji-Hirano-M]

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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?

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ABJM Matrix Model

N₁=N₂=N Z(N) =

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ABJM Matrix Model

[ ∏ sinh ∏ sinh / ∏ cosh ]²

N₁=N₂=N Z(N) =

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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

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Hidden Structure as Fermi Gas

After Some Calculation, ... [Marino-Putrov]

• Non-Interacting Fermi Gas

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

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Statistical Mechanics

• Canonical Ensemble: Sum in Conjugacy Class

⇒ Grand Canonical Ensemble

• Grand Potential

e^J^(μ) = 1 + ∑_N=1^∞Z^(N) e^-μN

• Inversely

Z^(N) = (2πi)^-1

∫

^{dμ e}^J^(μ)^{- μN}

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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_kμ³/3 + B_kμ + A_k

J^np^(μ) = ∑_l=1^∞( α^(l) μ²+ 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

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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

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Sample

(For k=1)

Z(1) = 1/4 Z(2) = 1/16π Z(3) = (π-3)/2⁶π

Z(4) = (-π²+10)/2¹⁰π² Z(5) = (-9π²+20π+26)/2¹²π²

Z(6) = (36π³-121π²+78)/2¹⁴3²π³

Z(7) = (-75π³+193π²+174π-126)/2¹⁶3π³

Z(8) = (1053π⁴-2016π³-4148π²+876)/2²¹3²π⁴

Z(9) = (5517π⁴-13480π³-15348π²+8880π+4140)/2²³3²π⁴

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Method: Factorization

ρ^(q₁^,q₂⁾= E^(q₁⁾E^(q₂⁾[M ^(q₁⁾+ M^(q₂⁾]^-1 ρⁿ^(q₁^,q₂⁾ = Σ_m (-1)^m [ρ⇓ ^mE]^(q₁⁾[ρ^n-1-mE]^(q₂⁾

x [M^(q₁⁾- (-1)ⁿM^(q₂⁾]^-1

[Tracy-Widom]

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Method: Hankel Matrix

• Density Matrix ρ: Isospectral to Hankel Matrix

ρ ≈ =

• A Magic Formula For Hankel Matrix

ρ₀ 0 ρ₁ 0 ρ₂ 0 ・ 0 ρ₁ 0 ρ₂ 0 ρ₃ ・ ρ₁ 0 ρ₂ 0 ρ₃ 0 ・ 0 ρ₂ 0 ρ₃ 0 ρ₄ ・ ρ₂ 0 ρ₃ 0 ρ₄ 0 ・ 0 ρ₃ 0 ρ₄ 0 ρ₅ ・

・・・・・・・・

ρ₊ 0 0 ρ_-

det (1+zρ_-) / det (1-zρ₊) = [(1-zρ₊)^-1E₊]₀ / [E₊]₀

[Hatsuda-M-Okuyama]

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Fitting Grand Potential

J^(μ) = log[ 1 + ∑_N=1 ^Nmax Z^(N) e^μN ] Strategy: Plot & Fit

J^(μ) vs J^pert^(μ)

(J^(μ)-J^pert^(μ))/e^-4μ/k vs α₁μ²+β₁μ+γ₁

(J^(μ)-J^pert^(μ)-J^np(1)^(μ))/e^-8μ/k vs α₂μ²+β₂μ+γ₂

(J^(μ)-J^pert^(μ)-J^np(1)^(μ)-J^np(2)^(μ))/e^-12μ/k vs α₃μ²+β₃μ+γ₃

・・・

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Example: k=1

(J^(μ)-J^pert^(μ))/e^-4μ

(J^(μ)-J^pert^(μ)-J^np(1)^(μ))/e^-8μ J^(μ)

(J^(μ)-J^pert^(μ)-J^np(1)^(μ)-J^np(2)^(μ))/e^-12μ

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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.

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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_kμ² + B_k- 8/3k] e^-8μ/k + ...

• Hereafter, J^osc^(μ)Abbreviated

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Results from Fitting

J_k=1(μ) = [(4μ²+μ+1/4)/π²]e^-4μ + [-(52μ²+μ/2+9/16)/(2π²)+2]e^-8μ + [(736μ²-152μ/3+77/18)/(3π²)-32]e^-12μ + ...

J_k=2(μ) = [(4μ²+2μ+1)/π²]e^-2μ + [-(52μ²+μ+9/4)/(2π²)+2]e^-4μ + [(736μ²-304μ/3+154/9)/(3π²)-32]e^-6μ + ...

J_k=3^(μ) = [4/3]e^-4μ/3 + [-2]e^-8μ/3 + [(4μ²+μ+1/4)/(3π²)+20/9]e^-4μ + ...

J_k=4^(μ) = [1]e^-μ + [-(4μ²+2μ+1)/(2π²)]e^-2μ + [16/3]e^-3μ + ...

J_k=6(μ) = [4/3]e^-2μ/3 + [-2]e^-4μ/3 + [(4μ²+2μ+1)/(3π²)+20/9]e^-2μ + ...

up to 7-instanton

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Schematically

J_k=1(μ) = [#μ²+#μ+#]e^-4μ + [#μ²+#μ+#]e^-8μ + [#μ²+#μ+#]e^-12μ + ...

J_k=2^(μ) = [#μ²+#μ+#]e^-2μ + [#μ²+#μ+#]e^-4μ + [#μ²+#μ+#]e^-6μ + ...

J_k=3(μ) = [#]e^-4μ/3 + [#]e^-8μ/3 + [#μ²+#μ+#]e^-4μ + ...

J_k=4^(μ) = [#]e^-μ + [#μ²+#μ+#]e^-2μ + [#]e^-3μ + ...

...

J_k=6(μ) = [#]e^-2μ/3 + [#]e^-4μ/3 + [#μ²+#μ+#]e^-2μ + ...

WS(1) WS(2) WS(3)

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Worldsheet Instanton From Top String

Implication from Topological Strings J_k^WS^(μ) = ∑_m=1^∞d_k^(m) e^-4mμ/k Multi-Covering Structure

d_k^(m) = ∑_g ∑_n|m (-1)^m/n N^g_n/n (2 Sin[2πn/k])^2g-2 Gopakumar-Vafa Invariant on F₀=P¹xP¹

N^g_n

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Match with Topological String?

J_k=3(μ) = [#]e^-4μ/3 + [#]e^-8μ/3 + [#μ²+#μ+#]e^-4μ + ...

J_k=4^(μ) = [#]e^-μ + [#μ²+#μ+#]e^-2μ + [#]e^-3μ + ...

...

J_k=6(μ) = [#]e^-2μ/3 + [#]e^-4μ/3 + [#μ²+#μ+#]e^-2μ + ...

WS(1) WS(2) WS(3)

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Match with Topological String?

J_k=3(μ) = [#]e^-4μ/3 + [#]e^-8μ/3 + [#μ²+#μ+#]e^-4μ + ...

J_k=4^(μ) = [#]e^-μ + [#μ²+#μ+#]e^-2μ + [#]e^-3μ + ...

...

J_k=6(μ) = [#]e^-2μ/3 + [#]e^-4μ/3 + [#μ²+#μ+#]e^-2μ + ...

WS(1) WS(2) WS(3)

: Match : Divergent : Not-Match

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First Guess

Membrane Instanton & Worldsheet Instanton, Same Origin in M-theory

d_k^(m) e^-4mμ/k = (divergence) + [#μ²+#μ+#] e^-2lμ Around k = 2m/l

Correctly Speaking, ...

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Cancellation of Divergences?

J_k=3(μ) = [#]e^-4μ/3 + [#]e^-8μ/3 + [#μ²+#μ+#]e^-4μ + ...

J_k=4^(μ) = [#]e^-μ + [#μ²+#μ+#]e^-2μ + [#]e^-3μ + ...

...

J_k=6(μ) = [#]e^-2μ/3 + [#]e^-4μ/3 + [#μ²+#μ+#]e^-2μ + ...

WS(1) WS(2) WS(3)

MB(1) MB(2)

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Cancellation of Divergence

WS3

WS4 WS2

WS1

MB1 MB2 MB3 MB4

• Worldsheet m-Instanton [Sin 2πm/k]^-1

• Membrane l-Instanton [Sin πlk/2]^-1

k=1

k=6 k=2

k=5 k=3 k=4

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More Dynamical Figure

WS3

WS4 WS2

WS1

MB1 MB2 MB3 MB4

• Worldsheet m-Instanton [Sin 2πm/k]^-1

• Membrane l-Instanton [Sin πlk/2]^-1

k=1

k=6 k=2

k=5 k=3 k=4

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1-Membrane Instanton

• Vanishing in k=odd

• Canceling Divergence

• Matching the WKB data

a_k⁽¹⁾ = -4(π²k)^-1Cos[πk/2]

b_k⁽¹⁾ = 2π ^-1 Cot[πk/2] Cos[πk/2]

c_k⁽¹⁾ = ...

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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.

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2-Membrane Instanton

Cancellation of Divergence in (2,0)+(0,2m+1)

[Calvo-Marino]

a_k⁽²⁾ = ... b_k⁽²⁾ = ... c_k⁽²⁾ = ...

m

l k=1

k=3

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Bound States (1,m)?

Cancellation of Divergence in (2,0)+(1,m)+(0,2m) m

l k=2

k=4

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(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_k^(1,m)^(μ) = ... = a_k⁽¹⁾ d_k^(m) e^-2μ-4mμ/k

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More Bound States

Similarly,

J_k^(2,m)(μ) = (a_k⁽²⁾+(a_k⁽¹⁾)²/2) d_k^(m) e^-4μ-4mμ/k Match with (2,2)+(0,5) in k=3, ...

J_k^(3,m)(μ) = (a_k⁽³⁾+a_k⁽²⁾a_k⁽¹⁾+(a_k⁽¹⁾)³/6) d_k^(m) e^-6μ-4mμ/k

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All Bound States

• Finally

J_k^(l,m)(μ) = ∑ (a_k^(l¹⁾)ⁿ¹/(n₁)! ... (a_k^(l^L⁾)ⁿ^L /(n_L)!

x d_k^(m) e^-2lμ-4mμ/k

• Sum Over

n₁l₁ + ... + n_Ll_L = l

• ∑(a)ⁿ/(n)! ⇒ Exp[a] ??

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To Summarize

Originally

J(μ) = J^pert(μ) + J^MB(μ) + J^WS(μ) + J^bnd(μ)

J^pert(μ) = #μ³ + #μ + #

J^MB(μ) = ∑_l>0J^MB(l)(μ) = ∑_l>0(#μ² + #μ + #) e^-2lμ J^WS(μ) = ∑_m>0J^WS(m)(μ) = ∑_m>0 # e^-4mμ/k

J^bnd(μ) = ∑l>0,m>0 J^(l,m)(μ)

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Effective Chemical Potential

μ_eff = μ + # ∑_la_k^(l) 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

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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.

Non-perturbative Effects in ABJM Theory from Fermi Gas Approach