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

What is M-Theory?

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

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

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

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

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

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

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

N x M2 on R 8 / Z k [= R >0 x S 7 / Z k ] ABJM Theory [arXiv: ] U(N) -k U(N) k Gauge Field Bifundamental Matter Fields N=6 Chern-Simons-matter Theory

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

Recent Developments Partition Function Reduced to Matrix Model [Hama-Hosomichi-Lee, Jafferis] Free Energy in large N ⇒ F ~ N 3/2 [Drukker-Marino-Putrov] Perturbative Sum Z = Ai[N] (~ exp N 3/2 ) [Fuji-Hirano-M]

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

Results Definition [Grand Potential] exp J(μ) = ∑ 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]

All Explicitly In Topological Strings 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 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 τ1→τ,τ2→-τ F(T 1,T 2,τ 1,τ 2 ) NS Limit F NS (T 1,T 2,τ)=lim τ1→τ,τ2→0 2πiτ 2 F(T 1,T 2,τ 1,τ 2 )

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)

Interpretation for Divergences S 7 / Z k Worldsheet Inst 1-Instantonk-InstantonOff Fixed Pt cf: Twisted Sectors in String Orbifold

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

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

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

Other Interesting Contributions from YITP Honda... Constant Map Contribution Hanada... Large N Expansion Revisited Shigemori... Generalization for N 1 ≠ N 2 Ishiki... Similar Analysis in NS5-branes ・・・ Thank you for your attention.