1. Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions Talk at KEK for String Advanced Lectures, 2011.9.12 (Tue.) Jun Nishimura (KEK, Sokendai) Ref.) Kim-J.N.-Tsuchiya arXiv:1108.1540

3. Answers from a nonperturbative formulation of superstring theory in (9+1)-dimensions fundamental questions concerning our universe 1.Why (3+1)-dimensions ? 2.Why expanding ?

4. Previous proposal Euclideanized model with SO(10) symmetry SSB of SO(10) studied by Gaussian Expansion Method J.N.-Okubo-Sugino, arXiv:1108.1293 free energy of SO(d) symmetric vacua (d=2,3,4,5,6,7) minimum at d=3 extent of space-time finite in all directions motivated us to reconsider the formulation Ishibashi-Kawai-Kitazawa-Tsuchiya (’96)

5. Our new proposal : regularize SO(9,1) Lorentzian symmetric model without Wick rotation Monte Carlo simulation 3 out of 9 spatial directions start to expand at some “critical time”

6. Comments on related works Quantum Cosmology with minisuperspace approx. Causal Dynamical Traingulation Matrix Cosmology Freedman-Gibbons-Schnabl (’05) Matrix Big Bang Craps-Sethi-Verlinde (’05) Emergent gravity Steinacker (’11), Yang (’10) Vilenkin (’82,’84), Hartle-Hawking (’83) Ambjorn-Jurkiewicz-Loll (’05) We expect that what we are doing is essentially a first-principle calculation of the unified theory including quantum gravity.

7. Plan of the talk 1.Introduction 2.Matrix model for superstrings 3.Matrix model with SO(9,1) symmetry 4.Monte Carlo results 5.Mechanism of the SSB 6.Removing the cutoffs 7.Summary and discussions

9. IKKT model matrix regularization of the Green-Schwarz worldsheet action in the Schild gauge interactions between D-branes string field theory from SD eqs. for Wilson loops Ishibashi-Kawai-Kitazawa-Tsuchiya (’96) proposed as a nonperturbative definition of type IIB superstring theory in 10 dim. Fukuma-Kawai-Kitazawa-Tsuchiya (’98) c.f.) Matrix Theory Banks-Fischler-Shenker-Susskind (’96)

10. 10 Ｄ 4Ｄ4Ｄ 10 Hermitian matrices Derivation of low-energy effective theory branched-polymer-like system Aoki-Iso-Kawai-Kitazawa-Tada (’99) Explicit calculations by the Gaussian expansion method J.N.-Sugino (’02), J.N.-Okubo-Sugino, Kawai, Kawamoto, Kuroki, Matsuo, Shinohara, Aoyama, Shibusa,… Does our 4-dimensional space-time appear from 10-dimensions ?

11. Results of the Gaussian expansion method J.N.-Okubo-Sugino (arXiv:1108.1293) Minimum of the free energy occurs at d=3 Extent of space-time finite in all directions SSB of SO(10) : interesting dynamical property of IKKT model, but we still seem to miss some important ingredient… extended directions shrunken directions

13. Matrix model with SO(9,1) symmetry raised and lowered by the metric Hermitian matrices Wick rotation IKKT model SO(10) symmetry

14. Partition function of the Lorentzian model connection to the worldsheet theory opposite sign ! Unlike IKKT model, the path integral is ill-defined ! c.f.) complex in the case of IKKT model the phase factor induces SSB of SO(10) J.N.-Vernizzi (’00), Anagnostopoulos-J.N.(’02) Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01)

15. Regularizing the Lorentzian model In order to separate space and time, we “gauge fix” the boost invariance. (1) IR cutoff in the temporal direction SO(9) symmetry is still manifest.

16. Regularizing oscillating functions inserting unity convergence factor

17. Cure this divergence by imposing : (2) IR cutoff in the spatial direction

18. Thus we arrive at Yoneya ('97) Monte Carlo simulation : Rational Hybrid Monte Carlo algorithm no sign problem unlike in IKKT model

20. Extracting time evolution

22. “critical time” SSB

24. Consider a simpler problem : solution : representation matrices of a compact semi-simple Lie algebra with d generators Maximum is achieved for SU(2) algebra

26. (continuum limit) (infinite volume limit) The theory thus obtained has no parameters other than one scale paramter !

27. Clear large-N scaling behavior observed with (continuum limit)

28. The extent of time increases and the size of the universe becomes very large at later time. (infinite volume limit)

30. A new proposal for the nonperturbative formulation of type IIB superstring theory in ten dimensions. instead of making Wick rotation, we introduce the IR cutoffs for both temporal and spatial directions after “gauge fixing” the boost symmetry The two cutoffs can be removed in the large-N limit. The theory thus obtained has no parameters other than one scale parameter. Summary

31. Integrating over the scale factor first, we obtain a model without sign problem. c.f.) Monte Carlo studies of IKKT model difficult due to sign problem Monte Carlo simulation revealed SSB of SO(9) down to SO(3) at some critical time. mechanism is totally different from that for the SSB in the IKKT model the size of the 3d space increases with time Cosmological singularity is naturally avoided due to noncommutativity.

32. Crucial role played by SUSY c.f.) bosonic model no expansion and no SSB ! Future directions Does a local field theory on a commutative space-time appear at later time ? How do 4 fundamental interactions and the matter fields appear at later time ?