スライド 1

ABJM行列模型の最近の進展
森山翔文（名古屋大学KMI）
[arXiv:1106.4631]
with H.Fuji and S.Hirano
[arXiv:1207.4283, 1211.1251, 1301.5184]
with Y.Hatsuda and K.Okuyama
グラフで見るM2
森山翔文（名古屋大学KMI）
[arXiv:1106.4631]
with H.Fuji and S.Hirano
[arXiv:1207.4283, 1211.1251, 1301.5184]
with Y.Hatsuda and K.Okuyama
「M2、100人に聞きました」
修士論文が出来た時期？
結婚の予定？
10月
博士課程
11月
ポスドク
12月
1月
2月
まだ
それ以降
異性に興
味ない
"M2の分配関数"のグラフ
0
1
2
3
質問
二歳児：「なんでにょろにょろ？」
物理学者：「縦軸は？横軸は？」
Contents
1.
2.
3.
4.
5.
Motivation
Fermi Gas
Exact Results
NonPerturbative Effects
As Various Instantons
(Skipped)
6. Further Directions
Contents
1.
2.
3.
4.
5.
Motivation
Fermi Gas
Exact Results
NonPerturbative Effects
As Various Instantons
(Skipped)
6. Further Directions
Previously in Fuji-Hirano-M
Perturbative Terms of ABJM Matrix Model
N1=N2=N
Z(N) =
in 't Hooft Expansion Sum Up To ・・・
Previously in Fuji-Hirano-M
Z(N) =
(Up To Constant Maps & Instanton Effects)
cf: [Marino-Putrov, Honda et al]
• Airy Function
Ai(N) = (2πi)-1∫dμ exp[μ3/3 - μN]
• Renormalization of 't Hooft coupling λ = N/k
λren = λ - 1/24 + 1/(24k2)
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!!
Motivation① M2-brane
ABJ(M)
N=6 Chern-Simons Theory (N1,N2,k)
⇕
Min(N1,N2) M2 with |N1-N2| Fractional M2 on C4/Zk
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
Motivation① M2-brane
Partition Function of M2 WorldVolume Theory
N x M2
Z = Airy(N) ≈ exp[-N3/2]
DOF N3/2 Reproduced
[Drukker-Marino-Putrov]
Also Non-Perturbative
Corrections
Motivation① M2-brane
[Drukker-Marino-Putrov]
Non-Perturbative Corrections
• 't Hooft Expansion in Matrix Model
Exp[-2π√2N/k]
Identified as String Wrapping CP1 in CP3
• Asymptotic Expansion-like Analysis
Exp[-π√2Nk]
Identified as D2-brane Wrapping RP3 in CP3
Motivation② From Gaussian To ABJM
ABJM
CS
Super
CS q-deform
Superalg
Gauss
Dissatisfaction①
M-theory from 't Hooft Expansion F(N) = log[Z(N)]
F(N) = N-2[F0pert(λ) + e-√λ + e-2√λ + ...]
+ N0[F1pert(λ) + e-√λ + e-2√λ + ...]
+ N2[F2pert(λ) + e-√λ + e-2√λ + ...]
+ N3[F3pert(λ) + e-√λ + e-2√λ + ...]
N
+
......
M-theory Regime
k: Fixed
k
't Hooft Regime
λ=N/k: Fixed but Large
Dissatisfaction②
• Instanton Effects?
- Worldsheet Instantons?
- Membrane Instantons?
- Bound States?
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 CS? Cubic?
Membrane WorldVolume Theory
[Aharony-Bergman-Jafferis-Maldacena]
ZM(N) ~ ∫DA Exp S11SG
MTheory
ZCS(N) = ∫DA Exp ∫A dA- A∧A∧A
Chern-Simons
Theory
Wave Function of The Universe
[Ooguri-Verlinde-Vafa]
All Genus Partition Function
[Fuji-Hirano-M]
Airy
Function
Ai(N) = (2πi)-1∫dμ Exp[μ3/3 - μN]
Message from Airy③ Statistical Mechanics
Ai(N) = (2πi)-1∫dμ exp[μ3/3 - μN]
Grand 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?
Grand Potential, Simpler!?
Contents
1.
2.
3.
4.
5.
Motivation
Fermi Gas
Exact Results
NonPerturbative Effects
As Various Instantons
(Skipped)
6. Further Directions
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
Besides, Large N with k Fixed
Statistical Mechanics
Sum Over Permutations
σ ∊ SN
(-1)σ = (-1)#{Intersections}
12345
σ
12345
(1→2→3→)
Tr ρ3
(4→5→)
Tr ρ2
Statistical Mechanics
Sum Over Permutations
12345
σ ∊ SN
τ
⇓
Sum Over Conjugacy Classes
σ～ σ' ⇔ ∃τ σ' =
τ-1
σ τ
Characterized by Partition
m1
1
m2
2
m3
3
・・・
σ
τ -1
12345
(4→1→5→)
(2→3→)
(-1)σ' = (-1)σ : Independent Of Representative
Statistical Mechanics
Z(N) = (N!)-1 ∑σ (-1)σ ∫ ∏i dqi 〈qi|ρ|qσ(i)〉
⇓
Z(N) = (N!)-1 ∑'{mj} (phase) x (combi) x ∏j(Tr ρj)
• Sum ∑' : Sum under Constraint ∑j j mj = N
mj
• (combi) = N! / [∏j mj! j ]
(j-1)mj
• (phase) = ∏j (-1)
(j-1)mj
m
m
Z(N) = ∑'{mj} ∏j (-1)
(Tr ρj) j / [mj! j j]
mj
A Check
m
N! = ∑'{mj} N! / [∏j mj! j j]
⇕
mj
∞
1 = ∑'{mj} ∏j=1 1 / [mj! j ]
To Lift Constraint, Generating Function
mj
∞
N
∑N=0 ∑'{mj} ∏j=1 z / [mj! j ]
mj
∞
∞
j
= ∏j=1 ∑mj=0 (z /j) / mj!
= exp ∑j=1∞ zj /j
∞
= 1 / (1-z)
Grand Canonical Partition Function
• To Lift Constraint, Grand Partition Funcation
Ξ(z) = exp J(z) = ∑N=0 ∞ Z(N) zN
J(z) = - ∑j=1 ∞ Tr ρj (-z)j / j = Tr log (1 + z ρ)
Ξ(z) = det (1 + z ρ)
• Chemical Potential
z = eμ
• Back to Canonical Partition Function
Z(N) = ∳ dz/(2πi) eJ(z)/ zN+1
Approximation
• Thermodynamic Limit
ρ(E) = dn(E)/dE
J(μ) = ∫ dE ρ(E) log (1 + eμ-E)
• Saddle Point Approximation
exp F(N) = Z(N) ≈ ∫ dμ eJ(μ)-μN
N = ∂J/∂μ ⇒ μ ≈ μ*(N)
F(N) ≈ J(μ*) - μ*N
Monomial Behavior
• Thermodynamic Limit
n ≈ Es
ρ ≈ Es-1
J ≈ Lis+1(-eμ) ≈ μs+1
• Saddle Point Approximation
N ≈ μs
F ≈ μs+1 ≈ N(s+1)/s
Application to Fermi Gas
Fermi Gas
p
2E
log(2 cosh q/2) + log(2 cosh p/2) = E
2E
q
|q|+|p|= 2E
n(E) = 8E2/2πℏ = (2/π2k) E2
s=2
N3/2
WKB 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
• Quadratic Prefactor (Linearity in "log[2coshq/2] ~ q")
• (α(l), β(l), γ(l)) Determined in ℏ-Perturbation
Contents
1.
2.
3.
4.
5.
Motivation
Fermi Gas
Exact Results
NonPerturbative Effects
As Various Instantons
(Skipped)
6. Further Directions
World Records of Exact Values
• [Hatsuda-M-Okuyama 2012/07]
Nmax= 9 for
• [Putrov-Yamazaki 2012/07]
Nmax= 19 for
• [Hatsuda-M-Okuyama 2012/11]
Nmax= 44 for
Nmax= 20 for
Nmax= 18 for
Nmax= 16 for
Nmax= 14 for
k=1
k=1
k=1
k=2
k=3
k=4
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
[Tracy-Widom]
ρ(q1,q2) = E(q1) E(q2) [M (q1) + M(q2)]-1
⇓
ρn(q1,q2) = Σm (-1)m [ρmE](q1) [ρn-1-mE](q2)
x [M(q1) - (-1)n M(q2)]-1
Method: Hankel Matrix
[Hatsuda-M-Okuyama]
• Density Matrix ρ: Isospectral to Hankel Matrix
ρ≈
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-
• A Magic Formula For Hankel Matrix
det (1+zρ-) / det (1-zρ+) = [(1-zρ+)-1E+]0 / [E+]0
Contents
1.
2.
3.
4.
5.
Motivation
Fermi Gas
Exact Results
NonPerturbative Effects
As Various Instantons
(Skipped)
6. Further Directions
Strategy: Plot & Fit
Grand Potential
J(μ) = log[ 1 + ∑N=1Nmax Z(N) eμN ]
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
・・・
k=1
(J(μ)-Jpert(μ))/e-4μ
J(μ)
(J(μ)-Jpert(μ)-Jnp(1)(μ)-Jnp(2)(μ))/e-12μ
(J(μ)-Jpert(μ)-Jnp(1)(μ))/e-8μ
k=2
(J(μ)-Jpert(μ))/e-2μ
J(μ)
(J(μ)-Jpert(μ)-Jnp(1)(μ)-Jnp(2)(μ))/e-6μ
(J(μ)-Jpert(μ)-Jnp(1)(μ))/e-4μ
k=3
(J(μ)-Jpert(μ))/e-4μ/3
J(μ)
(J(μ)-Jpert(μ)-Jnp(1)(μ)-Jnp(2)(μ))/e-4μ
(J(μ)-Jpert(μ)-Jnp(1)(μ))/e-8μ/3
k=6
(J(μ)-Jpert(μ))/e-2μ/3
J(μ)
(J(μ)-Jpert(μ)-Jnp(1)(μ)-Jnp(2)(μ))/e-2μ
(J(μ)-Jpert(μ)-Jnp(1)(μ))/e-4μ/3
k=4
(J(μ)-Jpert(μ))/e-μ
J(μ)
(J(μ)-Jpert(μ)-Jnp(1)(μ)-Jnp(2)(μ))/e-3μ
(J(μ)-Jpert(μ)-Jnp(1)(μ))/e-2μ
Oscillatory Behavior!!
• Original Definition of Grand Potential
eJ(μ) = 1 + ∑N=1∞ Z(N) e-μN
Periodic in μ = μ + 2πi
• But, 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 + ...
No Oscillations in Partition Function
Back to Canonical Partition Function
Z(N) = ∳ dz/(2πi) eJ(z)/ zN+1
No Josc(μ) After Extending Integral Range
Z(N) = ∫-πi dμ/(2πi)
πi
eJ(μ)-μN
=
∫
∞
dμ/(2πi)
-∞
eJ
naive
(μ)-μN
Short Discussions
• General in All Models of Statistical Mechanics
- 2πi Periodicity in Grand Potential
- But Not in Its Perturbative Expansion
• Newly Found?
"グラフで見る大ポテンシャル"
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 + ...
(Josc(μ) Abbreviated)
up to 7-instanton
Contents
1.
2.
3.
4.
5.
Motivation
Fermi Gas
Exact Results
NonPerturbative Effects
As Various Instantons
(Skipped)
6. Further Directions
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)
: Match
WS(2)
: Divergent
WS(3)
: 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 + ...
-4μ + ...
Jk=3(μ) = [#]e-4μ/3
+ [#]e-8μ/3
+ [#μ2+#μ+#]eMB(2)
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)
Cancellation of Divergence
k=1
MB4
MB3 • Worldsheet m-Instanton
-1
[sin
2πm/k]
MB2
• Membrane l-Instanton
[sin πlk/2]-1
k=2
k=3
k=4
k=5
k=6
WS1
WS2
WS3
WS4MB1
More Dynamical Figure
k=1
MB4
MB3 • Worldsheet m-Instanton
-1
[sin
2πm/k]
MB2
• Membrane l-Instanton
[sin πlk/2]-1
k=2
k=3
k=4
k=5
k=6
WS1
WS2
WS3
WS4MB1
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]
m
k=3
k=1
l
ak(2) = ...
bk(2) = ...
ck(2) = ...
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
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.
2.
3.
4.
5.
Motivation
Fermi Gas
Exact Results
NonPerturbative Effects
As Various Instantons
(Skipped)
6. Further Directions
Further Direction [Work in Progress]
• Generality of Cancellation??
- More Examples
Wilson Loops? ABJ Extensions? ... ...
- General Matrix Model Terminology
Thank You For Your Attention.

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