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

2. 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

3. 「 M2 、 100 人に聞きました」

4. "M2 の分配関数 " のグラフ 1 2 0 3

5. 質問 二歳児： 「なんでにょろにょ ろ？」 物理学者： 「縦軸は？横軸は？」

6. Contents 1.Motivation 2.Fermi Gas 3.Exact Results 4.NonPerturbative Effects 5.As Various Instantons (Skipped) 6.Further Directions

7. Contents 1.Motivation 2.Fermi Gas 3.Exact Results 4.NonPerturbative Effects 5.As Various Instantons (Skipped) 6.Further Directions

8. Previously in Fuji-Hirano-M Perturbative Terms of ABJM Matrix Model in 't Hooft Expansion Sum Up To ・・・ Z(N) = N1=N2=NN1=N2=N

9. Previously in Fuji-Hirano-M (Up To Constant Maps & Instanton Effects) cf: [Marino-Putrov, Honda et al] Airy Function Ai(N) = (2πi) -1 ∫ dμ exp[μ 3 /3 - μN] Renormalization of 't Hooft coupling λ = N/k λ ren = λ - 1/24 + 1/(24k 2 ) Z(N) =

10. 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!!

11. Motivation① M2-brane 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 N =6 Chern-Simons Theory (N 1,N 2,k) ⇕ Min(N 1,N 2 ) M2 with |N 1 -N 2 | Fractional M2 on C 4 /Z k ABJ(M)

12. 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

13. Motivation① M2-brane Non-Perturbative Corrections 't Hooft Expansion in Matrix Model Exp[-2π√ 2N/k ] Identified as String Wrapping CP 1 in CP 3 Asymptotic Expansion-like Analysis Exp[-π√ 2Nk ] Identified as D2-brane Wrapping RP 3 in CP 3 [Drukker-Marino-Putrov]

14. Motivation② From Gaussian To ABJM ABJM CSSuper Gauss CS q-deformSuperalg

15. Dissatisfaction① M-theory from 't Hooft Expansion F(N) = log[Z(N)] F(N) = N -2 [F 0 pert (λ) + e -√ λ + e -2√ λ +...] + N 0 [F 1 pert (λ) + e -√ λ + e -2√ λ +...] + N 2 [F 2 pert (λ) + e -√ λ + e -2√ λ +...] + N 3 [F 3 pert (λ) + e -√ λ + e -2√ λ +...] +...... k N M-theory Regime k: Fixed 't Hooft Regime λ=N/k: Fixed but Large

16. Dissatisfaction② Instanton Effects? - Worldsheet Instantons? - Membrane Instantons? - Bound States?

17. 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?

18. Z CS (N) = ∫ DA Exp ∫ A dA- A ∧ A ∧ AZ M (N) ~ ∫ DA Exp S 11SG Ai(N) = (2πi) -1 ∫ dμ Exp[μ 3 /3 - μN] Message from Airy② CS? Cubic? 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]

19. Message from Airy③ Statistical Mechanics Ai(N) = (2πi) -1 ∫ dμ exp[μ 3 /3 - μN] 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? Grand Potential, Simpler!?

20. Contents 1.Motivation 2.Fermi Gas 3.Exact Results 4.NonPerturbative Effects 5.As Various Instantons (Skipped) 6.Further Directions

21. 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 Besides, Large N with k Fixed

22. Sum Over Permutations σ ∊ S N (-1) σ = (-1) #{Intersections} Statistical Mechanics σ 1 2 3 4 5 (1→2→3→) (4→5→) Tr ρ 3 Tr ρ 2

23. Sum Over Permutations σ ∊ S N ⇓ Sum Over Conjugacy Classes σ ～ σ' ⇔ ∃ τ σ' = τ - 1 σ τ Characterized by Partition 1 m 1 2 m 2 3 m 3 ・ ・ ・ (-1) σ' = (-1) σ : Independent Of Representative Statistical Mechanics σ τ -1 τ 1 2 3 4 5 (4→1→5→) (2→3→)

24. Statistical Mechanics Z(N) = (N!) -1 ∑ σ (-1) σ ∫ ∏ i dq i 〈 q i |ρ|q σ(i) 〉 ⇓ Z(N) = (N!) -1 ∑' {m j } (phase) x (combi) x ∏ j (Tr ρ j ) m j Sum ∑' : Sum under Constraint ∑ j j m j = N (combi) = N! / [∏ j m j ! j m j ] (phase) = ∏ j (-1) (j-1)m j Z(N) = ∑' {m j } ∏ j (-1) (j-1)m j (Tr ρ j ) m j / [m j ! j m j ]

25. A Check N! = ∑' {m j } N! / [∏ j m j ! j m j ] ⇕ 1 = ∑' {m j } ∏ j=1 ∞ 1 / [m j ! j m j ] To Lift Constraint, Generating Function ∑ N=0 ∞ ∑' {m j } ∏ j=1 ∞ z N / [m j ! j m j ] = ∏ j=1 ∞ ∑ m j =0 ∞ (z j /j) m j / m j ! = exp ∑ j=1 ∞ z j /j = 1 / (1-z)

26. Grand Canonical Partition Function To Lift Constraint, Grand Partition Funcation Ξ(z) = exp J(z) = ∑ N=0 ∞ Z(N) z N J(z) = - ∑ j=1 ∞ Tr ρ j (-z) j / j = Tr log (1 + z ρ) Ξ(z) = det (1 + z ρ) Chemical Potential z = e μ Back to Canonical Partition Function Z(N) = ∳ dz/(2πi) e J(z) / z N+1

27. Approximation Thermodynamic Limit ρ(E) = dn(E)/dE J(μ) = ∫ dE ρ(E) log (1 + e μ-E ) Saddle Point Approximation exp F(N) = Z(N) ≈ ∫ dμ e J(μ)-μN N = ∂J/∂μ ⇒ μ ≈ μ*(N) F(N) ≈ J(μ*) - μ*N

28. Monomial Behavior Thermodynamic Limit n ≈ E s ρ ≈ E s-1 J ≈ Li s+1 (-e μ ) ≈ μ s+1 Saddle Point Approximation N ≈ μ s F ≈ μ s+1 ≈ N (s+1)/s

29. Application to Fermi Gas Fermi Gas n(E) = 8E 2 /2π ℏ = (2/π 2 k) E 2 s = 2 2E2E 2E2E q p log(2 cosh q/2) + log(2 cosh p/2) = E |q|+|p|= 2E N 3/2

30. 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 Quadratic Prefactor (Linearity in "log[2coshq/2] ~ q" ) (α (l), β (l), γ (l) ) Determined in ℏ -Perturbation

31. Contents 1.Motivation 2.Fermi Gas 3.Exact Results 4.NonPerturbative Effects 5.As Various Instantons (Skipped) 6.Further Directions

32. 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

33. 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

34. Method: Factorization 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 [Tracy-Widom]

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

36. Contents 1.Motivation 2.Fermi Gas 3.Exact Results 4.NonPerturbative Effects 5.As Various Instantons (Skipped) 6.Further Directions

37. Strategy: Plot & Fit Grand Potential J (μ) = log[ 1 + ∑ N=1 N max Z (N) e μN ] 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 ・・・

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

39. k=2 (J (μ) -J pert (μ) )/e -2μ (J (μ) -J pert (μ) -J np(1) (μ) )/e -4μ J(μ)J(μ) (J (μ) -J pert (μ) -J np(1) (μ) -J np(2) (μ) )/e -6μ

40. k=3 (J (μ) -J pert (μ) )/e -4μ/3 (J (μ) -J pert (μ) -J np(1) (μ) )/e -8μ/3 J(μ)J(μ) (J (μ) -J pert (μ) -J np(1) (μ) -J np(2) (μ) )/e -4μ

41. k=6 (J (μ) -J pert (μ) )/e -2μ/3 (J (μ) -J pert (μ) -J np(1) (μ) )/e -4μ/3 J(μ)J(μ) (J (μ) -J pert (μ) -J np(1) (μ) -J np(2) (μ) )/e -2μ

42. k=4 (J (μ) -J pert (μ) )/e -μ (J (μ) -J pert (μ) -J np(1) (μ) )/e -2μ J(μ)J(μ) (J (μ) -J pert (μ) -J np(1) (μ) -J np(2) (μ) )/e -3μ

43. Oscillatory Behavior!! Original Definition of Grand Potential e J(μ) = 1 + ∑ N=1 ∞ Z (N) e -μN Periodic in μ = μ + 2πi But, No More in J pert (μ) etc.

44. 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 +...

45. No Oscillations in Partition Function Back to Canonical Partition Function Z(N) = ∳ dz/(2πi) e J(z) / z N+1 No J osc (μ) After Extending Integral Range Z(N) = ∫ -πi πi dμ/(2πi) e J(μ)-μN = ∫ -∞ ∞ dμ/(2πi) e J naive (μ)-μN

46. 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 "

47. 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 (J osc (μ) Abbreviated)

48. Contents 1.Motivation 2.Fermi Gas 3.Exact Results 4.NonPerturbative Effects 5.As Various Instantons (Skipped) 6.Further Directions

49. 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)

50. 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 F 0 =P 1 xP 1 N g n

51. 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)

52. 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

53. 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,...

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

55. 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

56. 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

57. 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) =...

58. 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.

59. 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

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

61. (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

62. More Bound States Similarly, (2,m) Bound States 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,... (3,m) Bound States 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

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

64. 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)

65. Effective Chemical Potential μ eff = μ + # ∑ l a 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

66. Contents 1.Motivation 2.Fermi Gas 3.Exact Results 4.NonPerturbative Effects 5.As Various Instantons (Skipped) 6.Further Directions

67. Further Direction [Work in Progress] Generality of Cancellation?? - More Examples Wilson Loops? ABJ Extensions?...... - General Matrix Model Terminology Thank You For Your Attention.