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M-theory & Matrix Models

M-theory & Matrix Models. Sanefumi Moriyama ( NagoyaU -KMI/ KyotoU -YITP). What is M-Theory?. M is for Mother. 5 Consistent String Theories in 10D. Het-E8xE8. IIA. Het-SO(32). IIB. I. M is for Mother. 5 Consistent String Theories in 10D. 5 Va cua of A Unique String Theory.

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M-theory & Matrix Models

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  1. M-theory & Matrix Models Sanefumi Moriyama (NagoyaU-KMI/KyotoU-YITP)

  2. What is M-Theory?

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

  4. 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

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

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

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

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

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

  10. ABJM Theory [arXiv: 0806.1218] N=6Chern-Simons-matter Theory U(N)k U(N)-k Gauge Field Gauge Field Bifundamental Matter Fields N x M2 on R8/ Zk[= R>0x S7 / Zk]

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

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

  13. Recent Developments (Cont'd) • Worldsheet Instanton (F1 wrapping CP1⊂CP3) [Drukker-Marino-Putrov] • Membrane Instanton (D2 wrapping RP3⊂CP3) [Drukker-Marino-Putrov, Hatsuda-M-Okuyama] • Bound State [Hatsuda-M-Okuyama]

  14. Results Definition [Grand Potential] exp J(μ)= ∑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 • J(μ)=Jpert(μeff)+JWS(μeff)+J'MB(μeff) • Jpert(μ)=Cμ3/3+Bμ+A • JWS(μeff)=Ftop(T1eff,T2eff,λ) • J'MB(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 • T1eff=4μeff/k-iπ • T2eff=4μeff/k+iπ • λ=2/k • 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)

  16. 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) MB(2) MB(1)

  17. Interpretation for Divergences Worldsheet Inst S7 / Zk 1-Instanton k-Instanton Off Fixed Pt cf: Twisted Sectors in String Orbifold

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

  19. 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

  20. 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

  21. Other Interesting Contributions from YITP • Honda ... Constant Map Contribution • Hanada... Large N Expansion Revisited • Shigemori ... Generalization for N1 ≠ N2 • Ishiki... Similar Analysis in NS5-branes ・・・ Thank you for your attention.

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