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
in 't Hooft Expansion Sum Up To ・・・
Z(N) =
N1=N2=N
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) =
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!!
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
Special Case
No Fractional Branes: N1=N2=N Flat Space: k=1
IIA (C4/Zk ⇒ 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/Zk ABJ(M)
Motivation① M2-brane
Partition Function of M2 WorldVolume Theory
Z = Airy(N) ≈ Exp[-N3/2] DOF N3/2 Reproduced
[Drukker-Marino-Putrov]
N x M2
Also Non-Perturbative Corrections
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]
Motivation② From Gaussian To ABJM
ABJM
CS Super
Gauss
CS q-deform 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?
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]
Message from Airy③
Ai(N) = (2πi)-1
∫
dμ Exp[μ3/3 - μN]Chern-Simons Partition Function ?
ZCS(N) =
∫
DA Exp∫
A dA- A∧A∧AGrand Potential in Statistical Mechanics?
eJ(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
N1=N2=N Z(N) =
ABJM Matrix Model
[ ∏ sinh ∏ sinh / ∏ cosh ]2
N1=N2=N Z(N) =
Hidden Structure as Fermi Gas
1. Cauchy Determinant
∏sinh ∏sinh / ∏cosh = Det cosh-1
= ∑σ (-1)σ ∏i [2 cosh (μi-νσ(i))/2]-1
2. Trivialization of One Permutation
∑σ,τ (-1)σ+τ ∏i [2 cosh(μi-νσ(i))/2]-1 [2 cosh(μi-ντ(i))/2]-1
= N! ∑σ (-1)σ ∏i [2 cosh(μi-νi)/2]-1 [2 cosh(μi-νσ(i))/2]-1
3. Fourier Transform
(μ,ν) ⇒ (p,q)
4. (μ,ν) Gaussian Integration
Hidden Structure as Fermi Gas
After Some Calculation, ... [Marino-Putrov]
• Non-Interacting Fermi Gas
Z(N) = (N!)-1 ∑σ (-1)σ ∫ ∏i dqi 〈qi| ρ |qσ(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
N3/2 & Airy Easily(!) Reproduced
Statistical Mechanics
• Canonical Ensemble: Sum in Conjugacy Class
⇒ Grand Canonical Ensemble
• Grand Potential
eJ(μ) = 1 + ∑N=1∞ Z(N) e-μN
• Inversely
Z(N) = (2πi)-1
∫
dμ eJ(μ) - μNWKB Analysis
• ℏ(=2πk)-Perturbation
• Systematic Expansion in e-2μ ~ Exp[-π√2Nk] J(μ) = Jpert(μ) + Jnp(μ)
Jpert(μ) = Ck μ3/3 + Bk μ + Ak
Jnp(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
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]
Nmax= 9 for k=1
• [Putrov-Yamazaki 2012/07]
Nmax= 19 for k=1
• [Hatsuda-M-Okuyama 2012/11]
Nmax= 44 for k=1 Nmax= 20 for k=2 Nmax= 18 for k=3 Nmax= 16 for k=4 Nmax= 14 for k=6
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
Method: Factorization
ρ(q1,q2) = E(q1) E(q2) [M (q1) + M(q2)]-1 ρn(q1,q2) = Σm (-1)m [ρ⇓ mE](q1) [ρn-1-mE](q2)
x [M(q1) - (-1)n M(q2)]-1
[Tracy-Widom]
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]
Contents
1. Motivation 2. Fermi Gas
3. Exact Results
4. NonPerturbative Effects 5. Further Direction
Fitting Grand Potential
J(μ) = log[ 1 + ∑N=1 Nmax Z(N) eμN ] Strategy: Plot & Fit
J(μ) vs Jpert(μ)
(J(μ)-Jpert(μ))/e-4μ/k vs α1μ2+β1μ+γ1
(J(μ)-Jpert(μ)-Jnp(1)(μ))/e-8μ/k vs α2μ2+β2μ+γ2
(J(μ)-Jpert(μ)-Jnp(1)(μ)-Jnp(2)(μ))/e-12μ/k vs α3μ2+β3μ+γ3
・・・
Example: k=1
(J(μ)-Jpert(μ))/e-4μ
(J(μ)-Jpert(μ)-Jnp(1)(μ))/e-8μ J(μ)
(J(μ)-Jpert(μ)-Jnp(1)(μ)-Jnp(2)(μ))/e-12μ
Oscillatory Behavior!!
• Grand Potential
• Original Definition
eJ(μ) = 1 + ∑N=1∞ Z(N) e-μN Periodic in μ = μ + 2πi
• No More in Jpert(μ) etc.
To Remedy the 2πi-Periodicity
• 2πi-Periodic Grand Potential
Exp[J(μ)] = ∑N=-∞∞ Exp[Jnaive(μ+2πiN)] Jnaive(μ) = Jpert(μ) + Jnp(μ)
J(μ) = Jnaive(μ) + Josc(μ)
• Results:
Josc(μ) = 2 Cos[Ck μ2 + Bk - 8/3k] e-8μ/k + ...
• Hereafter, Josc(μ) Abbreviated
Results from Fitting
Jk=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 + ...
Jk=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 + ...
Jk=3(μ) = [4/3]e-4μ/3 + [-2]e-8μ/3 + [(4μ2+μ+1/4)/(3π2)+20/9]e-4μ + ...
Jk=4(μ) = [1]e-μ + [-(4μ2+2μ+1)/(2π2)]e-2μ + [16/3]e-3μ + ...
Jk=6(μ) = [4/3]e-2μ/3 + [-2]e-4μ/3 + [(4μ2+2μ+1)/(3π2)+20/9]e-2μ + ...
up to 7-instanton
Schematically
Jk=1(μ) = [#μ2+#μ+#]e-4μ + [#μ2+#μ+#]e-8μ + [#μ2+#μ+#]e-12μ + ...
Jk=2(μ) = [#μ2+#μ+#]e-2μ + [#μ2+#μ+#]e-4μ + [#μ2+#μ+#]e-6μ + ...
Jk=3(μ) = [#]e-4μ/3 + [#]e-8μ/3 + [#μ2+#μ+#]e-4μ + ...
Jk=4(μ) = [#]e-μ + [#μ2+#μ+#]e-2μ + [#]e-3μ + ...
...
Jk=6(μ) = [#]e-2μ/3 + [#]e-4μ/3 + [#μ2+#μ+#]e-2μ + ...
WS(1) WS(2) WS(3)
Worldsheet Instanton From Top String
Implication from Topological Strings JkWS(μ) = ∑m=1∞ dk(m) e-4mμ/k Multi-Covering Structure
dk(m) = ∑g ∑n|m (-1)m/n Ngn/n (2 Sin[2πn/k])2g-2 Gopakumar-Vafa Invariant on F0=P1xP1
Ngn
Match with Topological String?
Jk=1(μ) = [#μ2+#μ+#]e-4μ + [#μ2+#μ+#]e-8μ + [#μ2+#μ+#]e-12μ + ...
Jk=2(μ) = [#μ2+#μ+#]e-2μ + [#μ2+#μ+#]e-4μ + [#μ2+#μ+#]e-6μ + ...
Jk=3(μ) = [#]e-4μ/3 + [#]e-8μ/3 + [#μ2+#μ+#]e-4μ + ...
Jk=4(μ) = [#]e-μ + [#μ2+#μ+#]e-2μ + [#]e-3μ + ...
...
Jk=6(μ) = [#]e-2μ/3 + [#]e-4μ/3 + [#μ2+#μ+#]e-2μ + ...
WS(1) WS(2) WS(3)
Match with Topological String?
Jk=1(μ) = [#μ2+#μ+#]e-4μ + [#μ2+#μ+#]e-8μ + [#μ2+#μ+#]e-12μ + ...
Jk=2(μ) = [#μ2+#μ+#]e-2μ + [#μ2+#μ+#]e-4μ + [#μ2+#μ+#]e-6μ + ...
Jk=3(μ) = [#]e-4μ/3 + [#]e-8μ/3 + [#μ2+#μ+#]e-4μ + ...
Jk=4(μ) = [#]e-μ + [#μ2+#μ+#]e-2μ + [#]e-3μ + ...
...
Jk=6(μ) = [#]e-2μ/3 + [#]e-4μ/3 + [#μ2+#μ+#]e-2μ + ...
WS(1) WS(2) WS(3)
: Match : Divergent : Not-Match
First Guess
Membrane Instanton & Worldsheet Instanton, Same Origin in M-theory
dk(m) e-4mμ/k = (divergence) + [#μ2+#μ+#] e-2lμ Around k = 2m/l
Correctly Speaking, ...
Cancellation of Divergences?
Jk=1(μ) = [#μ2+#μ+#]e-4μ + [#μ2+#μ+#]e-8μ + [#μ2+#μ+#]e-12μ + ...
Jk=2(μ) = [#μ2+#μ+#]e-2μ + [#μ2+#μ+#]e-4μ + [#μ2+#μ+#]e-6μ + ...
Jk=3(μ) = [#]e-4μ/3 + [#]e-8μ/3 + [#μ2+#μ+#]e-4μ + ...
Jk=4(μ) = [#]e-μ + [#μ2+#μ+#]e-2μ + [#]e-3μ + ...
...
Jk=6(μ) = [#]e-2μ/3 + [#]e-4μ/3 + [#μ2+#μ+#]e-2μ + ...
WS(1) WS(2) WS(3)
MB(1) MB(2)
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
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
1-Membrane Instanton
• Vanishing in k=odd
• Canceling Divergence
• Matching the WKB data
ak(1) = -4(π2k)-1 Cos[πk/2]
bk(1) = 2π -1 Cot[πk/2] Cos[πk/2]
ck(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)
[Calvo-Marino]
ak(2) = ... bk(2) = ... ck(2) = ...
m
l k=1
k=3
Bound States (1,m)?
Cancellation of Divergence in (2,0)+(1,m)+(0,2m) m
l k=2
k=4
(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
Jk(1,m)(μ) = ... = ak(1) dk(m) e-2μ-4mμ/k
More Bound States
Similarly,
• (2,m) Bound States
Jk(2,m)(μ) = (ak(2)+(ak(1))2/2) dk(m) e-4μ-4mμ/k Match with (2,2)+(0,5) in k=3, ...
• (3,m) Bound States
Jk(3,m)(μ) = (ak(3)+ak(2)ak(1)+(ak(1))3/6) dk(m) e-6μ-4mμ/k
All Bound States
• Finally
Jk(l,m)(μ) = ∑ (ak(l1))n1/(n1)! ... (ak(lL))nL /(nL)!
x dk(m) e-2lμ-4mμ/k
• Sum Over
n1 l1 + ... + nL lL = l
• ∑(a)n/(n)! ⇒ Exp[a] ??
To Summarize
Originally
J(μ) = Jpert(μ) + JMB(μ) + JWS(μ) + Jbnd(μ)
Jpert(μ) = #μ3 + #μ + #
JMB(μ) = ∑l>0 JMB(l)(μ) = ∑l>0 (#μ2 + #μ + #) e-2lμ JWS(μ) = ∑m>0 JWS(m)(μ) = ∑m>0 # e-4mμ/k
Jbnd(μ) = ∑l>0,m>0 J(l,m)(μ)
Effective Chemical Potential
μeff = μ + # ∑l ak(l) e-2lμ
• Grand Potential
J(μ) = Jpert(μeff) + J'MB(μeff) + JWS(μeff) J'MB(μeff) = ∑l>0 (#μeff + #) e-2lμeff
• 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.