1. Summing Up All Genus Free Energy of ABJM Matrix Model Sanefumi Moriyama (Nagoya U) JHEP [arXiv:1106.4631] with H.Fuji and S.Hirano

2. M is for Mother IIA IIB I Het-SO(32) Het-E8xE8 5 Consistent String Theories in 10D

3. M is for Mother IIA IIB I Het-SO(32) Het-E8xE8 5 Consistent String Theories in 10D T-duality S-dualityOrientifold 5 Vacua of A Unique String Theory

4. M is for Mother M (11D) IIA IIB I Het-SO(32) Het-E8xE8 M2 / M5 10D & Compactifications Non- Perturbative Strong Coupling Limit

5. M is for Mystery 11D Supergrav as LowEnergyTheory Supergrav Sol for M2- & M5-branes Near Horizon Geometry Superconformal Symmetry osp(8|4) Action for Single M2-/M5-brane DOF N 3/2 /N 3 for N M2-/M5-branes Little was known before ABJM

6. cf. D-branes DOF N 2 for D-branes MatrixDescribed by

7. M is for Membrane Non-Abelian M2-brane theory by ABJM [Aharony-Bergman-Jafferis-Maldacena] Partition Function Localized to CS Matrix [Kapustin-Willett-Yaakov, Hama-Hosomichi-Lee] Planar N 3/2 Behavior Reproduced [Drukker-Marino-Putrov] Today: All Genus Sum A breakthrough by ABJM

8. Summary of Strategy & Result Chern-Simons Theory = Open A-model Duality: Closed B-model Holomorphic Anomaly Eq (Recurrence Eq) Modified Bessel Differential Eq Solution Implications?

9. Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

10. Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

11. (N 1 +N 2 )/2 M2 with N 1 -N 2 fractional M2 on C 4 / Z k ABJ(M) theory (N 1 =N 2 ) U(N 2 ) -k U(N 1 ) k Gauge Field Bifundamental Matter Fields N=6 Chern-Simons-matter Theory Superpotential A 1, A 2 B 1, B 2

12. N=2 Chern-Simons Theory Dim Red of 4D N=1 N=2 Vector Multiplet V N=2 Chiral Multiplet Φ L gauge (V) =... L matter (Φ) =... (Topological & Auxiliary)

13. N=3 Chern-Simons-matter Theory Field Contents of N=4 N=4 Vector Multiplet (V, φ) N=4 HyperMultiplet (Φ i, Φ i † ) Adj Rep Conj Rep (For General Gauge Groups & Representations) (CS Term: N=3 / No Kinetic Term for φ)

14. N=6 Chern-Simons-matter Theory Application to U(N 1 ) x U(N 2 ) U(N 2 ) -k U(N 1 ) k A 1, A 2 B 1, B 2

15. N=6 Chern-Simons-matter Theory so(3) = su(2) diag ⊂ su(2) A x su(2) B ⊂ su(4) = so(6) A 1, A 2 B 1, B 2 su(2) R-Symmetry

16. Brane Construction NS5 0 1 2 3 4 5 (1,k)5 0 1 2 [3,7] θ [4,8] θ [5,9] θ tanθ = k D3 0 1 2 6 N 1 x D3 N 2 x D3 (1,k)5 NS5 Preserving N=6 SUSY (Expected to be N=6 Chern-Simons-matter Theory)

17. Brane Construction T 6 -duality KKm (6dual) + k D6 KKm (6dual) (N 1 +N 2 )/2 x D2 (N 1 -N 2 ) x fractional D2 KKm (6dual, k x 10) KKm (6dual) (N 1 +N 2 )/2 x M2 (N 1 -N 2 ) x fractional M2 C 4 / Z k M-lift

18. Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

19. Localization Grassmann-odd Symmetry Grassmann-odd Quantity Integration Localized to

20. Grassmann-odd Symmetry δ: Chiral SUSY Grassmann-odd Quantity V: δV = S YM Localized to F μν = 0 & D μ σ = 0 (σ = A 4 ) Pure Gauge A μ = 0 & Constant σ Classical Action: Quadratic If Rescaled by t 1-Loop Exact Application to ABJM

21. Gauge Field Matter Field... = cosh -2 Application to ABJM

22. Finally, Localized to Matrix Model

23. Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

24. ABJM Matrix Model

25. If, instead Lens Space Matrix Model

26. If, further simplified Chern-Simons Matrix Model

27. The simplest one Gaussian Matrix Model Study from Gaussian MM to ABJM MM

28. Matrix Model Eigenvalue Density Wanted Definition - Resolvent - Planar Limit Eigenvalues

29. Nice Properties of Resolvent 1. Behavior 2.Dispersion Relation 3.EOM ・・・ Discontinuity Eq Integration Force

30. Integration Contour z Partition Func F 0 (λ)Resolvent ω 0 (z)

31. Chern-Simons Matrix Model

32. Resolvent Asymptotic Behavior as Discontinuity Eq from EOM

33. Chern-Simons Matrix Model A Regular Function [Halmagyi-Yasnov] Asymptotic Behavior Determined!

34. Lens Space Matrix Model

35. Same Tech as Chern-Simons Matrix Model Two Cuts Instead Lens Space Matrix Model sinh cosh

36. ABJM Matrix Model

37. Analytic Cont from Lens Space Matrix Model Result (for ABJM Slice: λ = λ 1 = λ 2 )

38. Result Neglecting Worldsheet Instanton N 3/2 Reproduced and More

39. Interpretation Charge Shift [Aharony-Hashimoto-Hirano-Ouyang] from Euler Coupling Match in Planar Case

40. Furthermore, Non-Planar Prediction Renormalization of 't Hooft coupling Or in terms of

41. Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

42. Strategy 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!!

43. Holomorphic Anomaly Eq General Case [Bershadsky-Cecotti-Ooguri-Vafa] Determined by F 0 : 2 nd & 3 rd Derivatives

44. Application to ABJM One Modulus = Ordinary Differential Eq 2 Cuts = Torus

45. Derivative 2 nd Derivative 3 rd Derivative (Yukawa Coupling) Elliptic Int Torus Modulus K: α-cycle K': β-cycle τ 1

46. Ansatz & Reduction Ansatz for Partition Function Eisenstein Series Reduced HAE [Drukker-Marino-Putrov] Covariant Derivative

47. Higher Genus Behavior

49. Resum 1 Discrepancy Discussed Later

50. Resum 2 After Partial Sum Recursion Relation (cf. SW theory [Huang-Klemm]) Translated into Differential Eq Solved by Modified Bessel Functions

51. Airy Function Result Result [Fuji-Hirano-M] Airy Function

52. Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

53. Discrepancy? SU(N) x SU(N) instead of U(N) x U(N)? Another Superconformal Theory by ABJM SU(N) instead of U(N) for D3-branes

54. Airy Function Modified Bessel Functions Everywhere as for Loop Operator VEV [Moore-Seiberg-Staudacher, Drukker-Gross] Cubic Integral Reminiscent of Chern-Simons Airy Function in Other Setups of M theory "Wave Function of the Universe" [Ooguri-Vafa-Verlinde] Corrections to Quantum Gravity? Max SUSY ・・・ No Corrections?

55. Future Directions [work in progress] Relation to Superalgebra U(N|N) Airy Function Beyond ABJM Slice Other Analysis (Surgery?) for WS Instanton Good Variables for ABJM Theory Three-Algebra? [Bagger-Lambert, Gutsvasson]