Exploring M-Theory and Matrix Models for String Theory Enthusiasts

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Delve into the intriguing world of M-Theory and Matrix Models with discussions on the different aspects including M-Theory's purpose, the Moduli Space of String Theory, Consistent String Theories, and the significance of D-branes in String Theory. Explore the mysteries and recent developments surrounding M-Theory in the realm of theoretical physics.

  • M-Theory
  • String Theory
  • Matrix Models
  • Theoretical Physics
  • Moduli Space
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  1. 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]

  2. M is NOT for Messier Catalogue M-Theory with Sym Enhancement M2 M5 We Are Here! Moduli Space of String Theory

  3. What is M-Theory?

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

  5. M is for Mother 5 Consistent String Theories in 10D 5 Vacua of A Unique String Theory Het-E8xE8 IIA String Duality D-brane Het-SO(32) IIB I

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

  7. M is for Membrane Fundamental M2-brane Solitonic M5-brane D2-brane String (F1) Lessons String Theory NOT Just "a theory of strings" Only Safe and Sound with D-branes

  8. M is for Mystery DOF N2for N D-branes Described by Matrix

  9. M is for Mystery M2-brane M2-brane DOF N3/2/N3for N M2-/M5-branes

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

  11. ABJM Theory [Aharony, Bergman, Jefferis, Maldacena] N=6 Chern-Simons-matter Theory U(N)k U(N)-k Gauge Field Gauge Field Bifundamental Matter Fields N x M2 on R8 / Zk

  12. 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]

  13. Recent Developments (Cont'd) Worldsheet Instanton (F1 wrapping CP1 CP3) [Drukker-Marino-Putrov, Hatsuda-M-Okuyama] Membrane Instanton (D2 wrapping RP3 CP3) [Drukker-Marino-Putrov, Hatsuda-M-Okuyama] Bound State [Hatsuda-M-Okuyama] (Basically From Numerical Studies)

  14. Results Def [Grand Potential] J( ) = log N=0 Z(N) e N Regarding Partition Function with U(N) x U(N) as PF of N-Particle Fermi Gas System [Marino-Putrov]

  15. All Explicitly In Topological Strings [Fuji-Hirano-M, (Hatsuda-M-Okuyama)3, Hatsuda-M-Marino-Okuyama] J( )=Jpert( eff)+JWS( eff)+JMB( eff) Jpert( )=C 3/3+B +A JWS( eff)=Ftop(T1eff,T2eff, ) JMB( eff)=(2 i)-1 [ FNS(T1eff/ ,T2eff/ ,1/ )] T1eff=4 eff/k-i T2eff=4 eff/k+i =2/k Ftop(T1,T2, ) = ... FNS(T1,T2, ) = ... C=2/ 2k, B=..., A=... eff = -(-1)k/22e-2 4F3(1,1,3/2,3/2;2,2,2;(-1)k/216e-2 ) +e-4 4F3(1,1,3/2,3/2;2,2,2;-16e-4 ) k=even k=odd

  16. All Explicitly In Topological Strings [Fuji-Hirano-M, (Hatsuda-M-Okuyama)3, Hatsuda-M-Marino-Okuyama] J( )=Jpert( eff)+JWS( eff)+JMB( eff) Jpert( )=C 3/3+B +A JWS( eff)=Ftop(T1eff,T2eff, ) JMB( eff)=(2 i)-1 [ FNS(T1eff/ ,T2eff/ ,1/ )] F(T1,T2, 1, 2): Free Energy of Refined Top Strings T1,T2: Kahler Moduli 1, 2: Coupling Constants Topological Limit Ftop(T1,T2, ) = lim 1 , 2 - F(T1,T2, 1, 2) NS Limit FNS(T1,T2, ) = lim 1 , 2 0 2 i 2F(T1,T2, 1, 2)

  17. All Explicitly In Topological Strings [Fuji-Hirano-M, (Hatsuda-M-Okuyama)3, Hatsuda-M-Marino-Okuyama] J( )=Jpert( eff)+JWS( eff)+JMB( eff) Jpert( )=C 3/3+B +A JWS( eff)=Ftop(T1eff,T2eff, ) JMB( eff)=(2 i)-1 [ FNS(T1eff/ ,T2eff/ ,1/ )] d1,d2 F(T1,T2, 1, 2) = jL,jR n d1,d2NjL,jR jL(qL) jR(qR) e-n(d1T1+d2T2) /[n(q1n/2-q1-n/2)(q2n/2-q2-n/2)] q1=e2 i 1q2=e2 i 2qL=e i( 1- 2)qR=e i( 1+ 2) d1,d2: BPS Index on local P1x P1 NjL,jR (Gopakumar-Vafa or Gromov-Witten invariants)

  18. Why Interesting? Non-Perturbative Part of Grand Potential J( ) Jk=1( ) = [# 2+# +#]e-4 + [# 2+# +#]e-8 + [# 2+# +#]e-12 + ... Jk=2( ) = [# 2+# +#]e-2 + [# 2+# +#]e-4 + [# 2+# +#]e-6 + ... Jk=3( ) = [#]e-4 /3 + [#]e-8 /3 Jk=4( ) = [#]e- + [# 2+# +#]e-2 + [#]e-3 ... Jk=6( ) = [#]e-2 /3 + [#]e-4 /3 + [# 2+# +#]e-4 + ... + ... + [# 2+# +#]e-2 + ...

  19. Why Interesting? Non-Perturbative Part of Grand Potential J( ) Jk=1( ) = [# 2+# +#]e-4 + [# 2+# +#]e-8 + [# 2+# +#]e-12 + ... Jk=2( ) = [# 2+# +#]e-2 + [# 2+# +#]e-4 + [# 2+# +#]e-6 + ... Jk=3( ) = [#]e-4 /3 + [#]e-8 /3 Jk=4( ) = [#]e- + [# 2+# +#]e-2 + [#]e-3 ... Jk=6( ) = [#]e-2 /3 + [#]e-4 /3 WS(1) WS(2) WS(3) + [# 2+# +#]e-4 + ... + ... + [# 2+# +#]e-2 + ...

  20. Why Interesting? Worldsheet Instanton Jk=1( ) = [# 2+# +#]e-4 + [# 2+# +#]e-8 + [# 2+# +#]e-12 + ... Jk=2( ) = [# 2+# +#]e-2 + [# 2+# +#]e-4 + [# 2+# +#]e-6 + ... Jk=3( ) = [#]e-4 /3 + [#]e-8 /3 Jk=4( ) = [#]e- + [# 2+# +#]e-2 + [#]e-3 ... Jk=6( ) = [#]e-2 /3 + [#]e-4 /3 WS(1) WS(2) WS(3) + [# 2+# +#]e-4 + ... + ... Match well with Topological String Prediction of WS + [# 2+# +#]e-2 + ...

  21. Why Interesting? Worldsheet Instanton, Divergent at Certain k Jk=1( ) = [# 2+# +#]e-4 + [# 2+# +#]e-8 + [# 2+# +#]e-12 + ... Jk=2( ) = [# 2+# +#]e-2 + [# 2+# +#]e-4 + [# 2+# +#]e-6 + ... Jk=3( ) = [#]e-4 /3 + [#]e-8 /3 Jk=4( ) = [#]e- + [# 2+# +#]e-2 + [#]e-3 ... Jk=6( ) = [#]e-2 /3 + [#]e-4 /3 WS(1) WS(2) WS(3) + [# 2+# +#]e-4 + ... + ... Match well with Topological String Prediction of WS + [# 2+# +#]e-2 + ...

  22. Why Interesting? Worldsheet Instanton, Divergent at Certain k Divergence Cancelled by Membrane Instanton Jk=1( ) = [# 2+# +#]e-4 + [# 2+# +#]e-8 + [# 2+# +#]e-12 + ... Jk=2( ) = [# 2+# +#]e-2 + [# 2+# +#]e-4 + [# 2+# +#]e-6 + ... Jk=3( ) = [#]e-4 /3 + [#]e-8 /3 Jk=4( ) = [#]e- + [# 2+# +#]e-2 + [#]e-3 ... Jk=6( ) = [#]e-2 /3 + [#]e-4 /3 WS(1) WS(2) WS(3) + [# 2+# +#]e-4 + ... MB(2) + ... Match well with Topological String Prediction of WS + [# 2+# +#]e-2 + ... MB(1)

  23. Divergence Cancellation Mechanism Aesthetically - Reproducing the Lessons String Theory, Not Just 'a theory of strings' Practically - Helpful in Determining Membrane Instanton

  24. Compact Moduli Space? Compactified by Membrane Instanton NonPerturbatively!? Perturbative WorldSheet Instanton Moduli

  25. Another Implication J( )=Jpert( eff)+JWS( eff)+JMB( eff) Jpert( )=C 3/3+B +A JWS( eff)=Ftop(T1eff,T2eff, ) JMB( eff)=(2 i)-1 [ FNS(T1eff/ ,T2eff/ ,1/ )] d1,d2 F(T1,T2, 1, 2) = jL,jR n d1,d2NjL,jR jL(qL) jR(qR) e-n(d1T1+d2T2) /[n(q1n/2-q1-n/2)(q2n/2-q2-n/2)] NonPerturbative Topological Strings on General Background by Requiring Divergence Cancellation [Hatsuda-Marino-M-Okuyama]

  26. Possible Because Viva! Max SUSY! ( Uniqueness, Solvability, Integrability) Assist from Numerical Studies Bound States, neither from 't Hooft genus-expansion nor from WKB -expansion

  27. 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

  28. ABJ Theory (N1N2) N=6 Chern-Simons-matter Theory U(N1)k U(N2)-k Gauge Field Gauge Field Bifundamental Matter Fields Min(N1,N2) x M2 & |N2-N1| x fractional M2 on R8 / Zk

  29. Fractional brane & Wilson loop WY k(N1,N2) One Point Function of Wilson Loop in Rep Y on Min(N1,N2) x M2 & |N2-N1| x fractional M2 Without Loss of Generality, M=N2-N1 0, k 0 [WY]GCk,M(z) = N=0 WY k(N,N+M) zN WY GCk,M(z) = [WY]GCk,M(z) / [1]GCk,0(z) ( [1]GCk,0(z) = exp J(log z) )

  30. Theorem [Hatsuda-Honda-M-Okuyama, Matsumoto-M] WY GCk,M(z) = det(M+r)x(M+r)Hp,q where (M = N2-N1) Elp( ) (1 + z Q( , ) P( , ) )-1 E-M+q-1( ) z Elp( ) (1 + z Q( , ) P( , ) )-1 Q( , )Eaq-M( ) (1 q-M r) (1 q M) Hp,q= and Q( , ) = [2cosh( - )/2]-1, P( , ) = [2cosh( - )/2]-1, Ej( ) = e(j+1/2) lp: p-th leg length aq: q-th arm length

  31. Theorem [Hatsuda-Honda-M-Okuyama, Matsumoto-M] WY GCk,M(z) = det(M+r)x(M+r)Hp,q where (M = N2-N1) Elp( )(1 + z Q( , ) P( , ) )-1 E-M+q-1( ) z Elp( ) (1 + z Q( , ) P( , ) )-1 Q( , )Eaq-M( ) (1 q-M r) (1 q M) Hp,q= and Q( , ) = ..., P( , ) = ..., Ej( ) = ... Q( , ) , P( , ) as Matrix, E( ) as Vector, Multiplication by Integration over , r? lp? aq?

  32. Theorem [Hatsuda-Honda-M-Okuyama, Matsumoto-M] WY GCk,M(z) = det(M+r)x(M+r)Hp,q where (M = N2-N1) Elp( )(1 + z Q( , ) P( , ) )-1 E-M+q-1( ) z Elp( )(1 + z Q( , ) P( , ) )-1 Q( , )Eaq-M( ) (1 q-M r) (1 q M) Hp,q= and Q( , ) = ..., P( , ) = ..., Ej( ) = e(j+1/2) lp: p-th leg length aq: q-th arm length

  33. Frobenius Symbol (a1a2ar|l1l2lr+M) [7,7,6,6,4,2,1] = [7,6,5,5,4,4,2]T U(N) x U(N) U(N) x U(N+3) (3,2,0|9,7,5,4,2,1) or (-1,-2,-3,3,2,0|9,7,5,4,2,1) (6,5,3,2|6,4,2,1)

  34. Example GC k,M=3 -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

  35. Especially, ABJM Wilson loop det " General Representation = det Hook Representations "

  36. Especially, ABJM Wilson loop " General Representation = det Hook Representations " " Solitonic Excitation = det Fundamental Excitation " Fundamental Excitation Hook Representation

  37. Especially, Fractional brane Fractional brane In terms of Wilson loop "Solitonic Branes from Fundamental Strings?" GC -1|#|2 -1|#|1 -1|#|0 -2|#|2 -2|#|1 -2|#|0 -3|#|2 -3|#|1 -3|#|0 det k,M=3

  38. 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

  39. Thank you for your attention!

  40. Pictorially S7 / Zk S7/ Zk k CP3x S1

  41. An Incorrect but Suggestive Interpretation Worldsheet Inst S7/ Zk 1-Instanton k-Instanton Off Fixed Pt cf: Twisted Sectors in String Orbifold

  42. Cancellation New Branch in WS inst Divergence Cancelled by MB Inst

  43. Compact Moduli Space Compactified by Membrane Instanton NonPerturbatively!? Perturbative WorldSheet Instanton Moduli Again: String Theory, NOT JUST "a theory of strings" Only Safe and Sound after D-branes

  44. Theorem [Hatsuda-Honda-M-Okuyama, Matsumoto-M] k(z) = Det (1 + z Q( , ) P( , ) ) WY GCk,M(z) / k(z) = det(M+r)x(M+r)Hp,q where Elp( ) (1 + z Q( , ) P( , ) )-1 E-M+q-1( ) z Elp( ) (1 + z Q( , ) P( , ) )-1 Q( , )Eaq-M( ) (1 q M) Hp,q= (1 q-M r) Q( , ) = [2cosh( - )/2]-1 P( , ) = [2cosh( - )/2]-1 Ej( ) = e(j+1/2) Q( , ), P( , ) as Matrix, E( ) as Vector, Multiplication by Integration over , r? lp? aq? (M = N2-N1)

  45. Frobenius Symbol For Young diagram [ 1 2 lmax] = [ '1 '2 'amax]T r = max{s| s-s-M 0} = max{s| 's-s+M 0}-M lp= 'p-p+M aq= q-q-M Denote as(a1a2 ar|l1l2 lr+M)

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