m-theory & matrix models sanefumi moriyama (nagoyau-kmi) [fuji+hirano+m 1106] [hatsuda+m+okuyama...
TRANSCRIPT
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(T1eff/λ,T2
eff/λ,1/λ)]
T1eff=4μeff/k-iπ
T2eff=4μeff/k+iπ
λ=2/k
μeff =μ-(-1)k/22e-2μ
4F3(1,1,3/2,3/2;2,2,2;(-1)k/216e-2μ)μ+e-4μ
4F3(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) = ...
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(T1eff/λ,T2
eff/λ,1/λ)]
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)
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(T1eff/λ,T2
eff/λ,1/λ)]
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=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)
Non-Perturbative Part of Grand Potential J(μ)
• Worldsheet Instanton
Why Interesting?
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
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
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)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)]
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(T1eff/λ,T2
eff/λ,1/λ)]
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
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)
r? lp? aq?
andQ(ν,μ) = ..., P(μ,ν) = ..., Ej(ν) = ...
(M = N2-N1)
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) = ..., 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
Theorem[Hatsuda-Honda-M-Okuyama, Matsumoto-M]
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 〉 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)
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)