1. M-Theory & Matrix Models Sanefumi Moriyama (NagoyaU-KMI) [Fuji+Hirano+M 1106] [Hatsuda+M+Okuyama 1207, 1211, 1301] [HMO+Marino 1306] [HMO+Honda 1306] [Matsumoto+M 1310]

2. M is NOT for Messier Catalogue We Are Here! Moduli Space of String Theory M-Theory with Sym Enhancement M2 M5

3. What is M-Theory?

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

5. M is for Mother IIA IIB I Het-SO(32) Het-E8xE8 5 Consistent String Theories in 10D 5 Vacua of A Unique String Theory String Duality D-brane

6. M is for Mother M (11D) IIA IIB I Het-SO(32) Het-E8xE8 10D Strong Coupling Limit

7. M is for Membrane Lessons String Theory NOT Just "a theory of strings" Only Safe and Sound with D-branes Fundamental M2-brane D2-brane String (F1) Solitonic M5-brane

8. M is for Mystery DOF N 2 for N D-branes MatrixDescribed by

9. M is for Mystery DOF N 3/2 /N 3 for N M2-/M5-branes M2-brane

10. To Summarize, we only know little on "What M-Theory Is" so far! Next, Recent Developments

11. N x M2 on R 8 / Z k ABJM Theory [Aharony, Bergman, Jefferis, Maldacena] U(N) -k U(N) k Gauge Field Bifundamental Matter Fields N=6 Chern-Simons-matter Theory

12. Recent Developments Partition Function Z(N) on S 3 ⇒ Matrix Model [Jafferis, Hama-Hosomichi-Lee] Free Energy F(N) = Log Z(N) in large N Limit F(N) ≈ N 3/2 [Drukker-Marino-Putrov] Perturbative Sum Z(N) = Ai[N] (≈ exp N 3/2 ) [Fuji-Hirano-M]

13. Recent Developments (Cont'd) Worldsheet Instanton (F1 wrapping CP 1 ⊂ CP 3 ) [Drukker-Marino-Putrov, Hatsuda-M-Okuyama] Membrane Instanton (D2 wrapping RP 3 ⊂ CP 3 ) [Drukker-Marino-Putrov, Hatsuda-M-Okuyama] Bound State [Hatsuda-M-Okuyama] (Basically From Numerical Studies)

14. Results Def [Grand Potential] J(μ) = log ∑ N=0 ∞ Z(N) e μN Regarding Partition Function with U(N) x U(N) as PF of N-Particle Fermi Gas System [Marino-Putrov]

15. All Explicitly In Topological Strings [Fuji-Hirano-M, (Hatsuda-M-Okuyama) 3, Hatsuda-M-Marino-Okuyama] J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=Cμ 3 /3+Bμ+A J WS (μ eff )=F top (T 1 eff,T 2 eff,λ) J MB (μ eff )=(2πi) -1 ∂ λ [λF NS (T 1 eff /λ,T 2 eff /λ,1/λ)] T 1 eff =4μ eff /k-iπ T 2 eff =4μ eff /k+iπ λ=2/k μ eff = μ-(-1) k/2 2e -2μ 4 F 3 (1,1,3/2,3/2;2,2,2;(-1) k/2 16e -2μ ) μ+e -4μ 4 F 3 (1,1,3/2,3/2;2,2,2;-16e -4μ ) k=even k=odd C=2/π 2 k, B=..., A=... F top (T 1,T 2,τ) =... F NS (T 1,T 2,τ) =...

16. All Explicitly In Topological Strings [Fuji-Hirano-M, (Hatsuda-M-Okuyama) 3, Hatsuda-M-Marino-Okuyama] J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=Cμ 3 /3+Bμ+A J WS (μ eff )=F top (T 1 eff,T 2 eff,λ) J MB (μ eff )=(2πi) -1 ∂ λ [λF NS (T 1 eff /λ,T 2 eff /λ,1/λ)] F(T 1,T 2,τ 1,τ 2 ): Free Energy of Refined Top Strings T 1,T 2 : Kahler Moduli τ 1,τ 2 : Coupling Constants Topological Limit F top (T 1,T 2,τ) = lim τ 1 →τ,τ 2 →-τ F(T 1,T 2,τ 1,τ 2 ) NS Limit F NS (T 1,T 2,τ) = lim τ 1 →τ,τ 2 →0 2πiτ 2 F(T 1,T 2,τ 1,τ 2 )

17. F(T 1,T 2,τ 1,τ 2 ) = ∑ j L,j R ∑ n ∑ d 1,d 2 N j L,j R d 1,d 2 χ j L (q L ) χ j R (q R ) e -n(d 1 T 1 +d 2 T 2 ) /[n(q 1 n/2 -q 1 -n/2 )(q 2 n/2 -q 2 -n/2 )] N j L,j R d 1,d 2 : BPS Index on local P 1 x P 1 (Gopakumar-Vafa or Gromov-Witten invariants) q 1 =e 2πiτ 1 q 2 =e 2πiτ 2 q L =e πi(τ 1 -τ 2 ) q R =e πi(τ 1 +τ 2 ) All Explicitly In Topological Strings [Fuji-Hirano-M, (Hatsuda-M-Okuyama) 3, Hatsuda-M-Marino-Okuyama] J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=Cμ 3 /3+Bμ+A J WS (μ eff )=F top (T 1 eff,T 2 eff,λ) J MB (μ eff )=(2πi) -1 ∂ λ [λF NS (T 1 eff /λ,T 2 eff /λ,1/λ)]

18. 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 +... Why Interesting? Non-Perturbative Part of Grand Potential J(μ)

19. Why Interesting? 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) Non-Perturbative Part of Grand Potential J(μ)

20. Worldsheet Instanton Why Interesting? 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 well with Topological String Prediction of WS

21. Why Interesting? Worldsheet Instanton, Divergent at Certain k 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 well with Topological String Prediction of WS

22. Why Interesting? Worldsheet Instanton, Divergent at Certain k Divergence Cancelled by Membrane Instanton 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) Match well with Topological String Prediction of WS

23. Divergence Cancellation Mechanism Aesthetically - Reproducing the Lessons String Theory, Not Just 'a theory of strings' Practically - Helpful in Determining Membrane Instanton

24. Compact Moduli Space? Perturbative WorldSheet Instanton Moduli Compactified by Membrane Instanton NonPerturbatively!?

25. Another Implication NonPerturbative Topological Strings on General Background by Requiring Divergence Cancellation [Hatsuda-Marino-M-Okuyama] F(T 1,T 2,τ 1,τ 2 ) = ∑ j L,j R ∑ n ∑ d 1,d 2 N j L,j R d 1,d 2 χ j L (q L ) χ j R (q R ) e -n(d 1 T 1 +d 2 T 2 ) /[n(q 1 n/2 -q 1 -n/2 )(q 2 n/2 -q 2 -n/2 )] J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=Cμ 3 /3+Bμ+A J WS (μ eff )=F top (T 1 eff,T 2 eff,λ) J MB (μ eff )=(2πi) -1 ∂ λ [λF NS (T 1 eff /λ,T 2 eff /λ,1/λ)]

26. Possible Because Viva! Max SUSY! (≈ Uniqueness, Solvability, Integrability) Assist from Numerical Studies Bound States, neither from 't Hooft genus-expansion nor from WKB ℏ -expansion

27. Break Summary So Far - Explicit Form of Membrane Instanton - Exact Large N Expansion of ABJM Partition Function - Divergence Cancellation - Moduli Space of Membrane? Hereafter - Fractional Membrane from Wilson Loop

28. Min(N 1,N 2 ) x M2 & |N 2 -N 1 | x fractional M2 on R 8 / Z k ABJ Theory (N 1 ≠N 2 ) U(N 2 ) -k U(N 1 ) k Gauge Field Bifundamental Matter Fields N=6 Chern-Simons-matter Theory

29. Fractional brane & Wilson loop One Point Function of Wilson Loop in Rep Y on Min(N 1,N 2 ) x M2 & |N 2 -N 1 | x fractional M2 [W Y ] GC k,M (z) = ∑ N=0 ∞ 〈 W Y 〉 k (N,N+M) z N Without Loss of Generality, M=N 2 -N 1 ≧ 0, k ＞ 0 〈 W Y 〉 GC k,M (z) = [W Y ] GC k,M (z) / [1] GC k,0 (z) 〈 W Y 〉 k (N 1,N 2 ) ( [1] GC k,0 (z) = exp J(log z) )

30. Theorem [Hatsuda-Honda-M-Okuyama, Matsumoto-M] H p,q = 〈 W Y 〉 GC k,M (z) = det (M+r)x(M+r) H p,q where (1 ≦ q ≦ M) E l p (ν) (1 + z Q (ν,μ) P (μ,ν) ) -1 E -M+q-1 (ν) z E l p (ν) (1 + z Q (ν,μ) P (μ,ν) ) -1 Q (ν,μ) E a q-M (μ) (1 ≦ q-M ≦ r) and Q (ν,μ) = [2cosh(ν-μ)/2] -1, P (μ,ν) = [2cosh(μ-ν)/2] -1, E j (ν) = e (j+1/2)ν (M = N 2 -N 1 ) l p : p-th leg length a q : q-th arm length

31. Q (ν,μ), P (μ,ν) as Matrix, E (ν) as Vector, Multiplication by Integration over μ, ν Theorem [Hatsuda-Honda-M-Okuyama, Matsumoto-M] H p,q = 〈 W Y 〉 GC k,M (z) = det (M+r)x(M+r) H p,q where (1 ≦ q ≦ M) E l p (ν) (1 + z Q (ν,μ) P (μ,ν) ) -1 E -M+q-1 (ν) z E l p (ν) (1 + z Q (ν,μ) P (μ,ν) ) -1 Q (ν,μ) E a q-M (μ) (1 ≦ q-M ≦ r) r? l p ? a q ? and Q (ν,μ) =..., P (μ,ν) =..., E j (ν) =... (M = N 2 -N 1 )

32. Theorem [Hatsuda-Honda-M-Okuyama, Matsumoto-M] H p,q = 〈 W Y 〉 GC k,M (z) = det (M+r)x(M+r) H p,q where (1 ≦ q ≦ M) E l p (ν) (1 + z Q (ν,μ) P (μ,ν) ) -1 E -M+q-1 (ν) z E l p (ν) (1 + z Q (ν,μ) P (μ,ν) ) -1 Q (ν,μ) E a q-M (μ) (1 ≦ q-M ≦ r) and Q (ν,μ) =..., P (μ,ν) =..., E j (ν) = e (j+1/2)ν (M = N 2 -N 1 ) l p : p-th leg length a q : q-th arm length

33. Frobenius Symbol (a 1 a 2 …a r |l 1 l 2 …l r+M ) (6,5,3,2|6,4,2,1) (3,2,0|9,7,5,4,2,1) or (-1,-2,-3,3,2,0|9,7,5,4,2,1) U(N) x U(N)U(N) x U(N+3) [7,7,6,6,4,2,1] = [7,6,5,5,4,4,2] T

34. Example 〈 -1|#|9 〉〈 -1|#|7 〉〈 -1|#|5 〉〈 -1|#|4 〉〈 - 1|#|2 〉〈 -1|#|1 〉 〈 -2|#|9 〉〈 -2|#|7 〉〈 -2|#|5 〉〈 -2|#|4 〉〈 - 2|#|2 〉〈 -2|#|1 〉 〈 -3|#|9 〉〈 -3|#|7 〉〈 -3|#|5 〉〈 -3|#|4 〉〈 - 3|#|2 〉〈 -3|#|1 〉 〈 3|#|9 〉 〈 3|#|7 〉 〈 3|#|5 〉 〈 3|#|4 〉 〈 3|#|2 〉 〈 3|#|1 〉 〈 2|#|9 〉 〈 2|#|7 〉 〈 2|#|5 〉 〈 2|#|4 〉 〈 2|#|2 〉 〈 2|#|1 〉 〈 0|#|9 〉 〈 0|#|7 〉 〈 0|#|5 〉 〈 0|#|4 〉 〈 0|#|2 〉 〈 0|#|1 〉 det GC k,M=3

35. Especially, ABJM Wilson loop det " 〈 General Representation 〉 = det 〈 Hook Representations 〉 "

36. Especially, ABJM Wilson loop Fundamental Excitation Hook Representation " 〈 Solitonic Excitation 〉 = det 〈 Fundamental Excitation 〉 " " 〈 General Representation 〉 = det 〈 Hook Representations 〉 "

37. Especially, Fractional brane Fractional brane In terms of Wilson loop "Solitonic Branes from Fundamental Strings?" GC k,M=3 〈 -1|#|2 〉〈 -1|#|1 〉〈 - 1|#|0 〉 〈 -2|#|2 〉〈 -2|#|1 〉〈 - 2|#|0 〉 〈 -3|#|2 〉〈 -3|#|1 〉〈 - 3|#|0 〉 det

38. Summary & Further Directions ABJM Partition Function - Exact Large N Expansion - Divergence Cancellation Fractional Membrane from Wilson Loop Generalization for M2 Orientifolds, Orbifolds, Ellipsoid/Squashed S 3 Implication of Cancellation for M5 Exploring Moduli Space of M-theory

39. Thank you for your attention!

40. Pictorially S7S7 S 7 / Z k CP 3 x S 1 k→∞k→∞/ Z k

41. An Incorrect but Suggestive Interpretation S 7 / Z k Worldsheet Inst 1-Instantonk-InstantonOff Fixed Pt cf: Twisted Sectors in String Orbifold

42. Cancellation New Branch in WS inst ≈ Divergence Cancelled by MB Inst

43. Compact Moduli Space Perturbative WorldSheet Instanton Moduli Compactified by Membrane Instanton NonPerturbatively!? Again: String Theory, NOT JUST "a theory of strings" Only Safe and Sound after D-branes

44. Q (ν,μ), P (μ,ν) as Matrix, E (ν) as Vector, Multiplication by Integration over μ, ν Theorem [Hatsuda-Honda-M-Okuyama, Matsumoto-M] Q (ν,μ) = [2cosh(ν-μ)/2] -1 P (μ,ν) = [2cosh(μ-ν)/2] -1 E j (ν) = e (j+1/2)ν H p,q = Ξ k (z) = Det (1 + z Q (ν,μ) P (μ,ν) ) 〈 W Y 〉 GC k,M (z) / Ξ k (z) = det (M+r)x(M+r) H p,q where E l p (ν) (1 + z Q (ν,μ) P (μ,ν) ) -1 E -M+q-1 (ν) (1 ≦ q ≦ M) z E l p (ν) (1 + z Q (ν,μ) P (μ,ν) ) -1 Q (ν,μ) E a q-M (μ) (1 ≦ q-M ≦ r) r? l p ? a q ? (M = N 2 -N 1 )

45. Frobenius Symbol r = max{s|λ s -s-M ≧ 0} = max{s|λ' s -s+M ≧ 0}-M l p = λ' p -p+M a q = λ q -q-M For Young diagram [λ 1 λ 2 …λ l max ] = [λ' 1 λ' 2 …λ' a max ] T Denote as(a 1 a 2 …a r |l 1 l 2 …l r+M )