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M-Theory & Matrix Models

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

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M-Theory & Matrix Models

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  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 M-Theorywith Sym Enhancement M2 M5 We Are Here! Moduli Space of String Theory

  3. What is M-Theory?

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

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

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

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

  8. M is for Mystery DOF N2 for N D-branes Described by Matrix

  9. M is for Mystery M2-brane M2-brane DOFN3/2/N3for NM2-/M5-branes

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

  11. ABJM Theory [Aharony, Bergman, Jefferis, Maldacena] N=6Chern-Simons-matter Theory U(N)k U(N)-k Gauge Field Gauge Field Bifundamental Matter Fields N x M2 on R8 / Zk

  12. Recent Developments • Partition Function Z(N) on S3⇒ Matrix Model [Jafferis, Hama-Hosomichi-Lee] • Free Energy F(N)= Log Z(N)in large N Limit F(N)≈N3/2 [Drukker-Marino-Putrov] • Perturbative Sum Z(N) = Ai[N] (≈exp N3/2) [Fuji-Hirano-M]

  13. Recent Developments (Cont'd) • Worldsheet Instanton (F1 wrapping CP1⊂CP3) [Drukker-Marino-Putrov, Hatsuda-M-Okuyama] • Membrane Instanton (D2 wrapping RP3⊂CP3) [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)eμN 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)=Jpert(μeff)+JWS(μeff)+JMB(μeff) • Jpert(μ)=Cμ3/3+Bμ+A • JWS(μeff)=Ftop(T1eff,T2eff,λ) • JMB(μeff)=(2πi)-1∂λ[λFNS(T1eff/λ,T2eff/λ,1/λ)] • T1eff=4μeff/k-iπ • T2eff=4μeff/k+iπ • λ=2/k • Ftop(T1,T2,τ)= ... • FNS(T1,T2,τ)= ... C=2/π2k, B=..., A=... μ-(-1)k/22e-2μ4F3(1,1,3/2,3/2;2,2,2;(-1)k/216e-2μ) k=even μeff = μ+e-4μ4F3(1,1,3/2,3/2;2,2,2;-16e-4μ) k=odd

  16. All Explicitly In Topological Strings[Fuji-Hirano-M, (Hatsuda-M-Okuyama)3, Hatsuda-M-Marino-Okuyama] • J(μ)=Jpert(μeff)+JWS(μeff)+JMB(μeff) • Jpert(μ)=Cμ3/3+Bμ+A • JWS(μeff)=Ftop(T1eff,T2eff,λ) • JMB(μeff)=(2πi)-1∂λ[λFNS(T1eff/λ,T2eff/λ,1/λ)] • F(T1,T2,τ1,τ2): Free Energy • of Refined Top Strings • T1,T2: KahlerModuli • τ1,τ2: Coupling Constants • Topological Limit Ftop(T1,T2,τ) = limτ1→τ,τ2→-τ F(T1,T2,τ1,τ2) • NS Limit FNS(T1,T2,τ) = limτ1→τ,τ2→0 2πiτ2F(T1,T2,τ1,τ2)

  17. All Explicitly In Topological Strings[Fuji-Hirano-M, (Hatsuda-M-Okuyama)3, Hatsuda-M-Marino-Okuyama] • J(μ)=Jpert(μeff)+JWS(μeff)+JMB(μeff) • Jpert(μ)=Cμ3/3+Bμ+A • JWS(μeff)=Ftop(T1eff,T2eff,λ) • JMB(μeff)=(2πi)-1∂λ[λFNS(T1eff/λ,T2eff/λ,1/λ)] F(T1,T2,τ1,τ2) = ∑jL,jR∑n∑d1,d2NjL,jRd1,d2 χjL(qL) χjR(qR) e-n(d1T1+d2T2) /[n(q1n/2-q1-n/2)(q2n/2-q2-n/2)] • q1 =e2πiτ1q2 =e2πiτ2qL=eπi(τ1-τ2)qR=eπi(τ1+τ2) NjL,jRd1,d2:BPS Index on local P1 x P1 (Gopakumar-Vafa orGromov-Witten invariants)

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

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

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

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

  22. Why Interesting? • Worldsheet Instanton, Divergent at Certain k • Divergence Cancelled by Membrane Instanton 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 well with Topological String Prediction of WS MB(2) MB(1)

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

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

  25. Another Implication • J(μ)=Jpert(μeff)+JWS(μeff)+JMB(μeff) • Jpert(μ)=Cμ3/3+Bμ+A • JWS(μeff)=Ftop(T1eff,T2eff,λ) • JMB(μeff)=(2πi)-1∂λ[λFNS(T1eff/λ,T2eff/λ,1/λ)] F(T1,T2,τ1,τ2) = ∑jL,jR∑n∑d1,d2NjL,jRd1,d2 χjL(qL) χjR(qR) e-n(d1T1+d2T2) /[n(q1n/2-q1-n/2)(q2n/2-q2-n/2)] NonPerturbative Topological Strings on General Background by Requiring Divergence Cancellation [Hatsuda-Marino-M-Okuyama]

  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. ABJ Theory (N1≠N2) N=6Chern-Simons-matter Theory U(N1)k U(N2)-k Gauge Field Gauge Field Bifundamental Matter Fields Min(N1,N2)x M2 & |N2-N1| x fractional M2 on R8 / Zk

  29. Fractional brane & Wilson loop 〈WY〉k(N1,N2) One Point Function of Wilson Loop in Rep Y on Min(N1,N2) x M2 & |N2-N1| x fractional M2 • Without Loss of Generality, M=N2-N1≧0, k＞0 • [WY]GCk,M(z) = ∑N=0∞〈WY〉k(N,N+M)zN 〈WY〉GCk,M(z) =[WY]GCk,M(z) / [1]GCk,0(z) • ( [1]GCk,0(z) = exp J(log z) )

  30. Theorem[Hatsuda-Honda-M-Okuyama, Matsumoto-M] • 〈WY〉GCk,M(z) = det(M+r)x(M+r)Hp,q where • (M = N2-N1) Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 E-M+q-1(ν) • (1≦q≦M) • Hp,q = z Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 Q(ν,μ)Eaq-M(μ) • (1≦q-M≦r) and Q(ν,μ) =[2cosh(ν-μ)/2]-1, P(μ,ν) = [2cosh(μ-ν)/2]-1, Ej(ν)= e(j+1/2)ν lp: p-th leg length aq: q-th arm length

  31. Theorem[Hatsuda-Honda-M-Okuyama, Matsumoto-M] • 〈WY〉GCk,M(z) = det(M+r)x(M+r)Hp,q where • (M = N2-N1) Elp(ν)(1 + z Q(ν,μ) P(μ,ν) )-1 E-M+q-1(ν) • (1≦q≦M) • Hp,q = z Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 Q(ν,μ)Eaq-M(μ) • (1≦q-M≦r) and Q(ν,μ) = ..., P(μ,ν) = ..., Ej(ν)= ... Q(ν,μ) , P(μ,ν)as Matrix, E(ν)as Vector, Multiplication by Integration over μ,ν r? lp? aq?

  32. Theorem[Hatsuda-Honda-M-Okuyama, Matsumoto-M] • 〈WY〉GCk,M(z) = det(M+r)x(M+r)Hp,q where • (M = N2-N1) Elp(ν)(1 + z Q(ν,μ) P(μ,ν) )-1 E-M+q-1(ν) • (1≦q≦M) • Hp,q = z Elp(ν)(1 + z Q(ν,μ) P(μ,ν) )-1 Q(ν,μ)Eaq-M(μ) • (1≦q-M≦r) and Q(ν,μ) =..., P(μ,ν) = ..., Ej(ν)= e(j+1/2)ν lp: p-th leg length aq: q-th arm length

  33. FrobeniusSymbol (a1a2…ar|l1l2…lr+M) [7,7,6,6,4,2,1] = [7,6,5,5,4,4,2]T U(N) x U(N) U(N) x U(N+3) (3,2,0|9,7,5,4,2,1) or (-1,-2,-3,3,2,0|9,7,5,4,2,1) (6,5,3,2|6,4,2,1)

  34. Example GC k,M=3 〈-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

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

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

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

  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 S3 • Implication of Cancellation for M5 • Exploring Moduli Space of M-theory

  39. Thank you for your attention!

  40. Pictorially S7 / Zk S7 / Zk k→∞ CP3 x S1

  41. An Incorrect but Suggestive Interpretation Worldsheet Inst S7 / Zk 1-Instanton k-Instanton Off Fixed Pt cf: Twisted Sectors in String Orbifold

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

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

  44. Theorem[Hatsuda-Honda-M-Okuyama, Matsumoto-M] • Ξk(z) = Det (1 + z Q(ν,μ) P(μ,ν) ) • 〈WY〉GCk,M(z) / Ξk(z) = det(M+r)x(M+r)Hp,q where Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 E-M+q-1(ν) • (1≦q≦M) • Hp,q = z Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 Q(ν,μ)Eaq-M(μ) • (1≦q-M≦r) Q(ν,μ) =[2cosh(ν-μ)/2]-1 P(μ,ν) = [2cosh(μ-ν)/2]-1 Ej(ν)= e(j+1/2)ν Q(ν,μ), P(μ,ν)as Matrix, E(ν)as Vector, Multiplication by Integration over μ,ν r? lp? aq? • (M = N2-N1)

  45. FrobeniusSymbol For Youngdiagram[λ1λ2…λlmax] = [λ'1λ'2…λ'amax]T r= max{s|λs-s-M≧0} = max{s|λ's-s+M≧0}-M lp= λ'p-p+M aq= λq-q-M Denote as(a1a2…ar|l1l2…lr+M)

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