1. Does 4d space-time emerge dynamically from the IKKT matrix model ? Numerical Approaches to AdS/CFT, Large N and Gravity, Sep 28-Oct 2, 2009, Imperial College, London Jun Nishimura (KEK Theory Center)

2. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 2  based on collaborations with Konstantinos Anagnostopoulos (National Technical University, Athens) Tatsumi Aoyama (KEK) Takehiro Azuma （ Setsunan U. ） Masanori Hanada (Weizmann Inst.) Toshiyuki Okubo (Meijo U.) Fumihiko Sugino (Okayama Inst.)

3. 0. Introduction

4. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 4 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, derivation of SM Comparing string theory to QCD

5. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 5 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, derivation of SM Comparing string theory to QCD

6. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 6 IKKT 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

7. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 7 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

8. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 8 Plan of the talk 0. Introduction 1. Complex fermion determinant 2. Gaussian expansion method 3. Monte Carlo studies of one-loop effective theory 4. Full Monte Carlo studies (preliminary) 5. Summary

9. 1.Complex fermion determinant

10. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 10 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

11. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 11 Remarkable properties of the phase J.N.-Vernizzi (’00) Stationarity of the phase increases for lower d This effect compensates the entropy loss for lower d !

12. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 12 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)

13. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 13 Cousins of the IKKT matrix model 6d IKKT model 4d IKKT model complex fermion determinant 10d IKKT model We mostly focus on this model for technical simplicity

14. 2. Gaussian expansion method

15. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 15 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

16. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 16 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

17. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 17 GEM applied to 6d IKKT model Gaussian action Aoyama-J.N.-Okubo, in prep. Various symmetry breaking patterns

18. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 18 Results of GEM for the 6d IKKT model Aoyama-J.N.-Okubo, in prep. Krauth-Nicolai- Staudacher (’98) magnify this region SO(5) SO(4) SO(3)

19. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 19 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 :

20. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 20 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 shrunk directions (cont ’ d) 0.18

21. 3. Monte Carlo studies of one-loop effective theory

22. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 22 One-loop effective theory treat them as small fluctuations and keep only quadratic terms valid when long distance behavior

23. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 23 Branched-polymer-like structure Aoki-Iso-Kawai-Kitazawa-Tada(’98) due to SUSY 0

24. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 24 One-loop IKKT model with No SSB of SO(10) symmetry Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)

25. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 25 One-loop 6d IKKT model with No SSB of SO(6) symmetry Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)

26. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 26 Distribution of the normalized eigenvalues  in the phase-quenched model  in the full model How to include the effects of the phase In order to see the net effects of the phase, we use the normalized variables: finite N effects Results in the phase-quenched model : represents the effects of the phase

27. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 27 The behavior of the correction factor no. of d.o.f. favours collapsed configs. Anagnostopoulos-J.N. (’02) one-loop 6d IKKT model

28. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 28 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

29. 4. Full Monte Carlo studies (preliminary)

30. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 30 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., in prog.

31. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 31 Determination of the peak position (at small x) Precise position of the peak can be obtained by more careful large-N extrapolation of

32. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 32 Comparison of the peak heights turned out to be negligible ! The information of comes in only through the determination of

33. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 33 Effects of the phase at As far as, the left peak dominates !

34. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 34 0.6 GEM result translates to Prediction from GEM consistent with MC !

35. 5. Summary

36. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 36 Summary and future prospects IKKT matrix model non-perturbative definition of superstring the dynamical origin of space-time dim. 6d IKKT model complex fermion determinant Gaussian expansion method MC studies with factorization method large-N extrapolation for each factor of the distribution so far, consistent with GEM prediction Extension to IKKT model, straightforward SO(6)SO(3) SSB

37. Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 37 Some comments D0 brane system Hanada and Wiseman’s talk The phase of the fermion determinant can be neglected. agreement with the gauge/gravity duality the reason needs to be understood SSB of SO(9) seems unlikely to happen… studies based on GEM Aoyama-J.N.-Okubo-Takeuchi, in prog IKKT model appearance of 4d gravity Kawai’s talk spontaneous breaking of SUSY ? Sugino’s talk How to think about the Euclidean time Ambjorn’s talk How to read off the Standard Model d.o.f.