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

We Are Here!

Moduli Spaceof String Theory

M-Theory with Sym EnhancementM2 M5

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

Fundamental M2-brane

D2-braneString(F1)

Solitonic M5-brane

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 only know little on"What M-Theory Is" so far!

Next, Recent Developments

N x M2 on R8 / Zk

ABJM Theory [Aharony, Bergman, Jefferis, Maldacena]

U(N)-kU(N)k

Gauge Field Gauge Field

Bifundamental Matter Fields

N=6 Chern-Simons-matter Theory

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]

Recent Developments (Cont'd)

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

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

• Bound State [Hatsuda-M-Okuyama]

(Basically From Numerical Studies)

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]

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,T2

eff,λ)JMB(μeff)=(2πi)-1∂λ[λFNS(T1

eff/λ,T2eff/λ,1/λ)]

T1eff=4μeff/k-iπ

T2eff=4μeff/k+iπ

λ=2/k

μeff = μ-(-1)k/22e-2μ4F3(1,1,3/2,3/2;2,2,2;(-1)k/216e-2μ)

μ+e-4μ4F3(1,1,3/2,3/2;2,2,2;-16e-4μ)

k=evenk=odd

C=2/π2k, B=..., A=...

Ftop(T1,T2,τ) = ...FNS(T1,T2,τ) = ...






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)

F(T1,T2,τ1,τ2) = ∑jL,jR∑n∑d1,d2

NjL,jRd1,d2

χjL(qL) χjR

(qR) e-n(d1T1+d2T2)

/[n(q1n/2-q1

-n/2)(q2n/2-q2

-n/2)]

NjL,jRd1,d2 : BPS Index on local P1 x P1

(Gopakumar-Vafa or Gromov-Witten invariants)

q1 =e2πiτ1 q2 =e2πiτ2 qL=eπi(τ1-τ2) qR=eπi(τ1+τ2)






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 + ...

Why Interesting?

Non-Perturbative Part of Grand Potential J(μ)

Why Interesting?


... Jk=6(μ) = [#]e-2μ/3 + [#]e-4μ/3 + [#μ2+#μ+#]e-2μ + ...

WS(1) WS(2) WS(3)

Non-Perturbative Part of Grand Potential J(μ)

• Worldsheet Instanton

Why Interesting?


... 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

Why Interesting?

• Worldsheet Instanton, Divergent at Certain k


... 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

Why Interesting?

• Worldsheet Instanton, Divergent at Certain k• Divergence Cancelled by Membrane Instanton


... Jk=6(μ) = [#]e-2μ/3 + [#]e-4μ/3 + [#μ2+#μ+#]e-2μ + ...

WS(1) WS(2) WS(3) MB(1)

MB(2)Match well with Topological String Prediction of WS

Divergence Cancellation Mechanism

• Aesthetically - Reproducing the Lessons

String Theory, Not Just 'a theory of strings'• Practically- Helpful in Determining Membrane Instanton

Compact Moduli Space?

Perturbative WorldSheet Instanton Moduli

Compactified by Membrane Instanton NonPerturbatively!?

Another Implication

NonPerturbative Topological Strings on General Background by Requiring Divergence Cancellation

[Hatsuda-Marino-M-Okuyama]

F(T1,T2,τ1,τ2) = ∑jL,jR∑n∑d1,d2

NjL,jRd1,d2

χjL(qL) χjR

(qR) e-n(d1T1+d2T2)

/[n(q1n/2-q1

-n/2)(q2n/2-q2

-n/2)]





Possible Because

• Viva! Max SUSY! (≈ Uniqueness, Solvability, Integrability)

• Assist from Numerical StudiesBound States,

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

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

Min(N1,N2) x M2 & |N2-N1| x fractional M2 on R8 / Zk

ABJ Theory (N1≠N2)

U(N2)-kU(N1)k

Gauge Field Gauge Field

Bifundamental Matter Fields

N=6 Chern-Simons-matter Theory

Fractional brane & Wilson loop

One Point Function of Wilson Loop in Rep Y on Min(N1,N2) x M2 & |N2-N1| x fractional M2

[WY]GCk,M(z) = ∑N=0

∞ 〈 WY 〉 k(N,N+M) zN

Without Loss of Generality, M=N2-N1 0, ≧ k ＞ 0

〈 WY 〉 GCk,M(z) = [WY]GC

k,M(z) / [1]GCk,0(z)

〈 WY 〉 k(N1,N2)

( [1]GCk,0(z) = exp J(log z) )

Theorem[Hatsuda-Honda-M-Okuyama, Matsumoto-M]

Hp,q =

〈 WY 〉 GCk,M(z) = det(M+r)x(M+r) Hp,q

where(1≦q≦M)Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 E-M+q-1(ν)

z Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 Q(ν,μ) Eaq-M(μ) (1≦q-M≦r)andQ(ν,μ) = [2cosh(ν-μ)/2]-1, P(μ,ν) = [2cosh(μ-ν)/2]-1, Ej(ν) = e(j+1/2)ν

(M = N2-N1)

lp: p-th leg length aq: q-th arm length

Q(ν,μ) , P(μ,ν) as Matrix, E(ν) as Vector,Multiplication by Integration over μ, ν


Hp,q =

〈 WY 〉 GCk,M (z) = det(M+r)x(M+r) Hp,q


z Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 Q(ν,μ) Eaq-M(μ) (1≦q-M≦r)

r? lp? aq?

andQ(ν,μ) = ..., P(μ,ν) = ..., Ej(ν) = ...

(M = N2-N1)


Hp,q =

〈 WY 〉 GCk,M (z) = det(M+r)x(M+r) 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

(M = N2-N1)

lp: p-th leg length aq: q-th arm length

Frobenius Symbol (a1a2…ar|l1l2…lr+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

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

Especially, ABJM Wilson loop

det

" 〈 General Representation 〉 = det 〈 Hook Representations 〉 "

Especially, ABJM Wilson loop

Fundamental Excitation

Hook Representation

" 〈 Solitonic Excitation 〉 = det 〈 Fundamental Excitation 〉 "

" 〈 General Representation 〉 = det 〈 Hook Representations 〉 "

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

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

Thank you for your attention!

Pictorially

S7

S7 / Zk

CP3 x S1

k→∞

/ Zk

An Incorrect but Suggestive Interpretation

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

Q(ν,μ), P(μ,ν) as Matrix, E(ν) as Vector,Multiplication by Integration over μ, ν


Q(ν,μ) = [2cosh(ν-μ)/2]-1

P(μ,ν) = [2cosh(μ-ν)/2]-1

Ej(ν) = e(j+1/2)ν

Hp,q =

Ξk(z) = Det (1 + z Q(ν,μ) P(μ,ν) )〈 WY 〉 GC

k,M (z) / Ξk(z) = det(M+r)x(M+r) Hp,qwhere

Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 E-M+q-1(ν) (1≦q≦M)

z Elp(ν) (1 + z Q(ν,μ) P(μ,ν) )-1 Q(ν,μ) Eaq-M(μ) (1≦q-M≦r)

r? lp? aq?(M = N2-N1)

Frobenius Symbol

r = max{s|λs-s-M 0} = max{≧ s|λ's-s+M 0}-≧ Mlp = λ'p-p+Maq = λq-q-M

For Young diagram [λ1λ2…λlmax] = [λ'1λ'2…λ'amax

]T

Denote as(a1a2…ar|l1l2…lr+M)

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