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Space-time picture suggested by the IIB matrix model

Space-time picture suggested by the IIB matrix model. YITP workshop “Discretization Approaches to the Dynanics of Space-time and Fields”, Sept.28, 2010 Jun Nishimura (KEK Theory Center & Graduate University for Advanced Studies). 0. Introduction.

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Space-time picture suggested by the IIB matrix model

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  1. Space-time picture suggested by the IIB matrix model YITP workshop “Discretization Approaches to the Dynanics of Space-time and Fields”, Sept.28, 2010 Jun Nishimura (KEK Theory Center & Graduate University for Advanced Studies)

  2. 0. Introduction

  3. Comparing string theory to QCD QCD string theory strong interactionswhat theory describesall the interactions including gravity free quarksperturbation theory10d space-time confinementnon-perturbative vacuum invisible extra dim. lattice theory non-perturbative formulation matrix models (Wilson ’74) (BFSS,IKKT ’96) properties of hadronsgoal black holes, early universe, SM and beyond Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  4. Comparing string theory to QCD QCD string theory strong interactionswhat theory describesall the interactions including gravity free quarksperturbation theory10d space-time confinementnon-perturbative vacuum invisible extra dim. lattice theory non-perturbative formulation matrix models (Wilson ’74) (BFSS,IKKT ’96) properties of hadronsgoal black holes, early universe, SM and beyond Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  5. IKKT matrix model (IIB matrix model) (Ishibashi-Kawai-Kitazawa-Tsuchiya ’96) a non-perturbative formulation of type IIB superstring theory in 10 dim. (conjecture) • Similarity to the Green-Schwarz worldsheet action in the Schild gauge • c.f.) Matrix Theory membrane action in the light cone gauge • Interactions between D-branes • Attempt to derive string field theory from SD eqs. for Wilson loops Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  6. in the limit Dynamical generation of 4d space-time c.f.) spontaneous breaking of Lorentz symmetry from tachyonic instability in bosonic SFT Kostelecky and Samuel (1988) The order parameter for the spontaneous breaking of the SO(10) symmetry “moment of inertia” tensor Eigenvalues : e.g.) SO(10) → SO(4) Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  7. Plan of the talk • 0. Introduction • Complex fermion determinant • 2. Gaussian expansion method • Aoyama-J.N.-Okubo,arXiv:1007.0883[hep-th] • 3. Monte Carlo studies (factorization method) • Anagnostopoulos-Azuma-J.N.,arXiv:1009.4504[cond-mat] • 4. Monte Carlo studies of 6d IKKT model (preliminary) • Anagnostopoulos-Aoyama-Azuma-Hanada-J.N., work in progress • 5. Summary Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  8. Complex fermion determinant

  9. Complex fermion determinant • fermion determinant • reweighting method simulate the phase quenched model complex in general cannot be treated as the Boltzmann weight suppressed as effective sampling becomes difficult “sign problem” Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  10. Remarkable properties of the phase J.N.-Vernizzi (’00) Stationarity of the phase increases for lower d This effect can compensate the entropy loss for lower d ! Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  11. This is a dilemma ! • Phase of the fermion determinant • important for the possible SSB of SO(10) • difficult to include in Monte Carlo simulation Gaussian expansion method Section 2 Sugino-J.N. (’00), Kawai et al. (’01),… New Monte Carlo technique Section 3 Anagnostopoulos-J.N. (’01),… Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  12. Models with similar properties (SSB of SO(D) expected due to complex fermion det.) 10d IKKT model 6d IKKT model 4d toy model (non SUSY) J.N. (’01) Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  13. 2. Gaussian expansion method

  14. Gaussian expansion method e.g.) one-matrix model Consider the Gaussian action free parameter Perform perturbative expansion using free propagator interaction vertex one-loop counterterm Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  15. Self-consistency equation Results of GEM depends on the free parameter e.g.) free energy of the one-matrix model plateau How to identify the plateau ? self-consistency eq.: Search for concentration of solutions Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  16. GEM applied to 6d IKKT model Aoyama-J.N.-Okubo, arXiv:1007.0883[hep-th] • 6d IKKT model • Gaussian action Various symmetry breaking patterns Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  17. magnify this region Results of GEM for the 6d IKKT model Aoyama-J.N.-Okubo, arXiv:1007.0883[hep-th] SO(5) SO(4) SO(3) SO(5) SO(4) SO(3) SO(5) SO(4) SO(3) Krauth-Nicolai- Staudacher (’98) Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  18. SO(6) SO(3) SSB Results of GEM for the 6d IKKT model (cont’d) concentration of solutions identified SO(5) SO(4) SO(5) SO(4) SO(3) SO(3) suggesting : Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  19. Results of GEM for the 6d IKKT model (cont’d) extent of the eigenvalue distribution in the extended/shrunk direction SO(3), extended SO(4), extended SO(5), extended Universal shrunken directions finite in units of Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  20. Constant-volume property Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  21. Understanding based on LEET treat them as small fluctuations and keep only quadratic terms branched-polymer-like structure (the reason for constant volume property) Aoki-Iso-Kawai-Kitazawa-Tada(’98) Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00) Shrunken directions dominated by the off-diagonal part SO(D) inv. typical scale of the branched polymer (the reason for the universal shrunken direction) Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  22. SO(2) ansatz Naively, disfavored. Gaussian action 13 free parameters Cyclic permutations of 4 free parameters Many solutions at order 5. d=3 is chosen dynamically in the 6d IKKT model Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  23. Reconsidering 10d IKKT model Sugino-J.N. (’00), Kawai et al. (’01),… Free energy is lower for d=4 than for d=7 (universal shrunken direction) Kawai –Kawamoto-Kuroki-Shinohara (’03) Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  24. Constant-volume property Consistent with preliminary MC data Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  25. Comparing SO(d) d=2,3,4,5,6,7in 10d IKKT model constant volume property ansatz order 1 order 3 SO(2) 6.49 3.05 SO(3) 7.0 -1.36 SO(4) 6.15 0.70 SO(5) 5.91 1.33 SO(6) 5.76 1.54 SO(7) 5.52 1.62 3.63[x2] 0.12[x6], 0.11, 0.05 3.24[x3] 0.10[x6], 0.08 1.35[x4] 0.14[x6] 0.84[x5] 0.11[x3], 0.11, 0.09 0.67[x6] 0.11[x3], 0.07 0.57[x7] 0.09[x3] universal shrunken direction Old results J.N.-Sugino (’02) New results (preliminary) J.N.-Okubo-Sugino, work in progress Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  26. Constant volume property in the 10D IKKT model Space-time picture suggested by the IIB matrix model

  27. 3. Monte Carlo studies by the factorization method

  28. The sign problem a general system cannot be treated as the Boltzmann weight reweighting method VEV w.r.t. phase-quenched model Exponentially large numbers of configurations are needed to achieve given accuracy. Space-time picture suggested by the IIB matrix model

  29. Moreover, there is also a general problem in the reweighting method Region of configuration space that gives important contribution to Region of configuration space sampled by simulating the phase-quenched model Overlap problem Space-time picture suggested by the IIB matrix model

  30. The basic idea of the factorization method Anagnostopoulos-J.N. (’02) Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat] Control some observables determine and sample effectively the important region of configuration space normalized observables Density of states Space-time picture suggested by the IIB matrix model

  31. Factorization property of the density of states reweighting formula constrained system Space-time picture suggested by the IIB matrix model

  32. The saddle-point equation effect of the phase (The constraints enable us to study the important regions.) Space-time picture suggested by the IIB matrix model

  33. Choice of observables Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat] the remaining overlap problem in evaluating constrained system Space-time picture suggested by the IIB matrix model

  34. Minimal set Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat] Assume that there is no more overlap problem with Saddle-point eq. In fact, there is no overlap problem with Space-time picture suggested by the IIB matrix model

  35. The role of the phase Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat] However, one can show that Note that Space-time picture suggested by the IIB matrix model

  36. A short summary of the method Choose the set of observables so that the remaining observables are (approximately) decorrelated with the phase; i.e., Space-time picture suggested by the IIB matrix model

  37. GEM results for the 4d toy model J.N. (’01) J.N.-Okubo-Sugino (’04) Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  38. The properties of the fermion determinant Integrating over fermionic variables, one obtains analogous to IKKT model ! Space-time picture suggested by the IIB matrix model

  39. Reproduce GEM results by the factorization method The result for the phase-quenched model Applying factorization method using , we have checked that the GEM results are indeed solutions to the saddle-point equations. Space-time picture suggested by the IIB matrix model

  40. Factorization method applied to the toy model Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat] 1.373(2) Similar agreement observed also for other equations. Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  41. Other possible dangerous observables… Remaining overlap problem is small. Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  42. 4. Monte Carlo studies of 6d IKKT model (preliminary) Anagnostopoulos-Aoyama-Azuma-Hanada-J.N. work in progress

  43. Let us recall some GEM results. constant volume property Universal shrunken directions We will see how these results can be reproduced by Monte Carlo simulation. Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  44. No SSB in the phase-quenched model 0.6 Space-time picture suggested by the IIB matrix model

  45. the normalized observables the phase-quenched model : finite N effects The use of the normalized variables enables us to see the net effects of the phase. Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  46. Factorization method Anagnostopoulos-J.N. (’02) Distribution of the normalized eigenvalues has a double-peak structure ! scales ! scales ! consistent with branched polymer L.h.s. is 1/N suppressed ! Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  47. Small x behavior of in full 6dIKKT model Anagnostopoulos-Aoyama-Azuma-Hanada-J.N., work in progress phase space suppression : Large-N extrapolation reveals the existence of a “hard-core potential” Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  48. Determination of the peak position (at small x) The extent of the hard core potential gives the (universal) shrunken direction. Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  49. Effects of the phase at The extent of the extended direction is almost decorrelated with the phase. No need to constrain the large eigenvalues. Constant volume property can be naturally understood. Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

  50. Comparison of the free energy e.g.) SO(2) and SO(3) almost negligible at large N The difference of the free energy density can be roughly determined by the difference of Jun Nishimura (KEK) 10.9.28 YITP workshop Space-time picture suggested by the IIB matrix model Space-time picture suggested by the IIB matrix model

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