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

2. M is for Mother M (11D) II-A II-B I Het-SO(32) Het-E8xE8 M2 / M5 10D & Compactifications Non- Perturbative Strong Coupling Limit

3. 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

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

5. M is for Matrix 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

6. Contents 1.Introduction 2.Planar Partition Function 3.All Genus Sum 4.Discussions

7. Contents 1.Introduction 2.Planar Partition Function 3.All Genus Sum 4.Discussions

8. ABJ(M) theory U(N 2 ) -k U(N 1 ) k Gauge Field Bifundamental Matter Fields Coincident M2-branes on S 7 / Z k Computation of Partition Function? N 1 =N 2 : No Fractional M2-branes (ABJM Slice)

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

10. Example 2 - 2 g = ∫ Mfd e -tδV = # (Poincare-Hopf Theorem)

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

12. Finally, Localized to Matrix Model

13. If, instead Lens Space Matrix Model

14. If, further simplified Chern-Simons Matrix Model

15. The simplest one Gaussian Matrix Model

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

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

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

19. Chern-Simons Matrix Model Resolvent Asymptotic Behavior as Discontinuity Eq from EOM

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

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

22. ABJM Matrix Model Analytic Cont from Lens Space Matrix Model Result (for ABJM Slice)

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

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

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

26. Contents 1.Introduction 2.Planar Partition Function 3.All Genus Sum 4.Discussions

27. Higher Genus Behavior from Duality 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)

28. Holomorphic Anomaly Eq General Case [Bershadsky-Cecotti-Ooguri-Vafa] Determined by F 0

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

30. Derivative 2 nd Derivative 3 rd Derivative (Yukawa Coupling) Elliptic F Torus Modulus

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

32. Higher Genus Behavior

33. Resum 1 Discrepancy Discussed Later

34. Resum 2 The Top Term Recursion Relation (cf. SW theory [Huang-Klemm]) Translated into Differential Eq Solved by Modified Bessel Functions

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

36. Contents 1.Introduction 2.Planar Partition Function 3.All Genus Sum 4.Discussions

37. Discrepancy? Lots of Conventional Factors? ex: Yukawa Coupling / 4 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

38. Airy Function Modified Bessel Functions Everywhere as for Loop Operator VEV [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?

39. Future Directions [work in progress] Relation to Superalgebra U(N|N) Airy Function Beyond ABJM Slice Other Analysis (Surgery?) for WS Instanton