Download - M-theory & Matrix Models
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M-theory & Matrix Models
Sanefumi Moriyama(NagoyaU-KMI/KyotoU-YITP)
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What is M-Theory?
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M is for Mother
IIA
IIB
I
Het-SO(32)
Het-E8xE8
5 Consistent String Theories in 10D
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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
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M is for Mother
M (11D)
IIA
IIB
I
Het-SO(32)
Het-E8xE8
10D
Strong Coupling Limit
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M is for Membrane
LessonsString Theory NOT Just "a theory of strings"
Only Safe and Sound with D-branes
M2-brane
D2-braneString(F1)
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M is for Mystery
DOF N2 for N D-branes
MatrixDescribed by
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M is for Mystery
DOF N3/2/N3 for N M2-/M5-branes
M2-braneM2-brane
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To Summarize, we know little on"What M-Theory Is" so far!
Next, Recent Developments
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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
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Pictorially
S7
S7 / Zk
CP3 x S1
k→∞
/ Zk
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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]
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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]
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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]
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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)
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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)
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Interpretation for Divergences
S7 / Zk
Worldsheet Inst
1-Instanton k-Instanton Off Fixed Pt
cf: Twisted Sectors in String Orbifold
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Cancellation
New Branch in WS inst ≈ Divergence
Cancelled by MB Inst
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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
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Possible Because
• Viva! Max SUSY! (≈ Uniqueness, Solvability, Integrability)
• Assist from Numerical StudiesBound States,
neither from 't Hooft genus-expansion nor from WKB -expansionℏ
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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.