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

StringDuality

D-brane

M is for Mother

M (11D)

IIA

IIB

I

Het-SO(32)

Het-E8xE8

10D

Strong Coupling Limit

M is for Membrane

LessonsString Theory NOT Just "a theory of strings"

Only Safe and Sound with D-branes

M2-brane

D2-braneString(F1)

M is for Mystery

DOF N2 for N D-branes

MatrixDescribed by

M is for Mystery

DOF N3/2/N3 for N M2-/M5-branes

M2-braneM2-brane

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

Next, Recent Developments

N x M2 on R8 / Zk [= R>0 x S7 / Zk]

ABJM Theory [arXiv: 0806.1218]

U(N)-kU(N)k

Gauge Field Gauge Field

Bifundamental Matter Fields

N=6 Chern-Simons-matter Theory

Pictorially

S7

S7 / Zk

CP3 x S1

k→∞

/ Zk

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]

Recent Developments (Cont'd)

• Worldsheet Instanton (F1 wrapping CP1 CP⊂ 3) [Drukker-Marino-Putrov]

• Membrane Instanton (D2 wrapping RP3 CP⊂ 3) [Drukker-Marino-Putrov, Hatsuda-M-Okuyama]

• Bound State [Hatsuda-M-Okuyama]

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]

All Explicitly In Topological Strings

J(μ)=Jpert(μeff)+JWS(μeff)+J'MB(μeff)Jpert(μ)=Cμ3/3+Bμ+A

JWS(μeff)=Ftop(T1eff,T2

eff,λ)J'MB(μeff)=(2πi)-1∂λ[λFNS(T1

eff/λ,T2eff/λ,1/λ)]

T1eff=4μeff/k-iπ

T2eff=4μeff/k+iπ

λ=2/k

F(T1,T2,τ1,τ2): Free Energy of Refined Top Strings T1,T2: Kahler Moduli 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)

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(1)

MB(2)

Interpretation for Divergences

S7 / Zk

Worldsheet Inst

1-Instanton k-Instanton Off 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 StudiesBound States,

neither from 't Hooft genus-expansion nor from WKB -expansionℏ

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.