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

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 3 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model QCD string theory strong interactions what theory describes all the interactions including gravity free quarks perturbation theory 10d space-time confinement non-perturbative vacuum invisible extra dim. lattice theory non-perturbative formulation matrix models (Wilson ’74) (BFSS,IKKT ’96) properties of hadrons goal black holes, early universe, SM and beyond Comparing string theory to QCD

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 4 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model QCD string theory strong interactions what theory describes all the interactions including gravity free quarks perturbation theory 10d space-time confinement non-perturbative vacuum invisible extra dim. lattice theory non-perturbative formulation matrix models (Wilson ’74) (BFSS,IKKT ’96) properties of hadrons goal black holes, early universe, SM and beyond Comparing string theory to QCD

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 5 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 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) YITP workshop Space-time picture suggested by the IIB matrix model 6 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model Dynamical generation of 4d space-time Eigenvalues : in the limit The order parameter for the spontaneous breaking of the SO(10) symmetry e.g.) SO(10) → SO(4) c.f.) spontaneous breaking of Lorentz symmetry from tachyonic instability in bosonic SFT Kostelecky and Samuel (1988) “moment of inertia” tensor

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 7 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model Plan of the talk 0. Introduction 1. Complex fermion determinant 2. Gaussian expansion method Aoyama-J.N.-Okubo, arXiv: [hep-th] 3. Monte Carlo studies (factorization method) Anagnostopoulos-Azuma-J.N., arXiv: [cond-mat] 4. Monte Carlo studies of 6d IKKT model (preliminary) Anagnostopoulos-Aoyama-Azuma-Hanada-J.N., work in progress 5. Summary

1.Complex fermion determinant

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

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 10 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 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) YITP workshop Space-time picture suggested by the IIB matrix model 11 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 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) YITP workshop Space-time picture suggested by the IIB matrix model 12 Models with similar properties 6d IKKT model 4d toy model (non SUSY) (SSB of SO(D) expected due to complex fermion det.) 10d IKKT model J.N. (’01) Space-time picture suggested by the IIB matrix model Jun Nishimura (KEK) YITP workshop

2. Gaussian expansion method

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

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

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

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

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

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

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

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 21 Understanding based on LEET treat them as small fluctuations and keep only quadratic terms Aoki-Iso-Kawai-Kitazawa-Tada(’98) Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00) branched-polymer-like structure (the reason for constant volume property) Space-time picture suggested by the IIB matrix model Jun Nishimura (KEK) YITP workshop 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) YITP workshop Space-time picture suggested by the IIB matrix model 22 SO(2) ansatz Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model d=3 is chosen dynamically in the 6d IKKT model 13 free parameters Gaussian action 4 free parameters Cyclic permutations of Naively, disfavored. Many solutions at order 5.

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

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

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 25 Comparing SO(d) d=2,3,4,5,6,7 in 10d IKKT model Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model order 1order 3 SO(2) SO(3) SO(4) SO(5) SO(6) SO(7) ansatz J.N.-Sugino (’02) New results (preliminary) J.N.-Okubo-Sugino, work in progress Old results 3.63[x2] 0.12[x6], 0.11, [x3] 0.10[x6], [x4] 0.14[x6] 0.84[x5] 0.11[x3], 0.11, [x6] 0.11[x3], [x7] 0.09[x3] universal shrunken direction constant volume property

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 26 Constant volume property in the 10D IKKT model

3. Monte Carlo studies by the factorization method

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

Jun Nishimura (KEK) YITP workshop 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 sampled by simulating the phase-quenched model Region of configuration space that gives important contribution to Overlap problem

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

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 31 Factorization property of the density of states reweighting formula constrained system

Jun Nishimura (KEK) YITP workshop 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.)

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

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

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

Jun Nishimura (KEK) YITP workshop 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.,

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

Jun Nishimura (KEK) YITP workshop 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 !

Jun Nishimura (KEK) YITP workshop 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.

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

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

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

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

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 44 No SSB in the phase-quenched model 0.6

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

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

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

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

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 49 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 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) YITP workshop Space-time picture suggested by the IIB matrix model 50 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model Comparison of the free energy almost negligible at large N The difference of the free energy density can be roughly determined by the difference of e.g.) SO(2) and SO(3)

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 51 Comparison of very subtle yet… More careful analysis will give us a definite conclusion.

5. Summary

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 53 Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model Summary and future prospects IKKT matrix model non-perturbative definition of superstring theory the dynamical origin of space-time dimensionality 6d IKKT model, 4d toy model complex fermion determinant, SSB of SO(D) expected Gaussian expansion method 4d toy : SO(4)SO(2) SSB 6d IKKT : SO(6)SO(d) SSB 10d IKKT : SO(10)SO(d) SSB universal extra dimension constant volume property true vacuum may be d=3…

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 54 Summary and future prospects (cont’d) MC studies of these models difficult due to the sign problem factorization method uses the factorization property of the density of states reduces the overlap problem by controlling observables extrapolations possible for the factorized functions the observables to be controlled have to be chosen appropriately for a general system demonstration in the 4d toy model 6d IKKT model universal shrunken directions constant volume property reproduced quantitatively !

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 55 Summary and future prospects (cont’d) Comparison of free energy for SO(d) vacua being pursued using GEM and MC GEM SO(2) should be studied more carefully comparison of SO(d) d=2,3,4,5,6,7 at the 5th order in 10d IKKT MC with factorization method so far, no evidence for disfavoring SO(2) large-N extrapolation is important A definite conclusion will be obtained soon.

Jun Nishimura (KEK) YITP workshop Space-time picture suggested by the IIB matrix model 56 Summary and future prospects (cont’d) Interpretation of IKKT model Branched polymer as low energy effective theory SUSY plays an important role in dynamically generating the notion of commutative space-time coordinates certain non-commutativity exists due to the off-diagonal elements “non-commutative extra dimension” Aschieri-Grammatikopoulos-Steinacker-Zoupanos The ratio R / r seems to be finite. d=3 may be chosen as the true vacuum. What does the IKKT model describe? The state of the early universe? How can we describe time evolution? Matrix Cosmology? Freedman-Schnabl-Gibbons