1. The origin of space-time as seen from matrix model simulations Seminar at KMI, Nagoya U., Nov. 8 (Tue.), 2011 Jun Nishimura (KEK,SOKENDAI) Ref.) M.Hanada, J.N., Y.Sekino, T.Yoneya, Phys.Rev. Lett. 104 (2010) 151601 arXiv: 1108.5153 S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540, 1110.4803

3. Quantum gravity Superstring theory is a natural candidate for a unified theory including quantum gravity Important problems in particle physics: the hierarchy problem why EW scale is much smaller than the Planck scale (or why gravity is so weak) the existence of dark energy, dark matter CMB, supernovae, structure formation, …

4. The testing ground for superstring theory 2 amazing predictions of Einstein’s general relativity Black hole Big bang singularities Quantum effects of gravity become crucial.

5. Important developments in the 90s Gauge-gravity duality (e.g., AdS/CFT correspondence) Maldacena (1997), Gubser-Klebanov-Polyakov, Witten (1998) Matrix model formulation of superstring/M theories Banks-Fischler-Shenker-Susskind (1996), Ishibashi-Kawai-Kitazawa-Tsuchiya (1997)  Gauge theory description of black hole thermodynamics  Correspondence at the level of local operators  Dynamical origin of space-time  Applications to the physics beyond the Standard Model Monte Carlo simulation provides an important tool to explore these two directions.

6. Plan of the talk 1.Introduction 2.Black hole thermodynamics from gauge theory 3. Direct test of gauge-gravity correspondence 4.(3+1)d expanding universe from matrix model c.f.) Tsuchiya’s talk on Oct.18 (Tue.) 5. Expanding universe as a classical solution 6. Summary and discussions

7. Hanada-J.N.-Takeuchi, PRL 99 (’07) 161602 Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08) 021601 Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09) 181602 Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602

8. N D0 branes t horizon black 0-brane solution in type IIA SUGRA 1d U(N) SUSY gauge theory near-extremal black hole at finite T In the decoupling limit, the D0 brane system describes the black hole microscopically. Itzhaki-Maldacena-Sonnenschein -Yankielowicz (’98) Gauge-gravity duality for D0-brane system SUGRA description : valid type IIA superstring

9. quantum description of the states inside the BH Gauge/gravity duality predicts that this should be reproduced by 1d SYM. large-N, low T microscopic origin of the black hole thermodynamics Prediction from gauge/gravity duality (I) dual geometry 7.41 Hawking’s theory black hole thermodynamics Klebanov-Tseytlin (’96)

10. Comparison including corrections corrections Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602 [arXiv:0811.3102]

11. M.Hanada, J.N., Y.Sekino, T.Yoneya : Phys.Rev. Lett. 104 (2010) 151601 arXiv: 1108.5153

12. Prediction from gauge/gravity duality (II) Gubser-Klebanov-Polyakov-Witten relation (’98) correlation functions in gauge theory generating functional operator-field correspondence gauge gravity SUGRA action evaluated at the classical solution with the boundary condition

13. Correlation functions in 1d SYM theory Perturbative calculations plagued by severe IR divergence : require genuinely non-perturbative methods 1) gauge-gravity correspondence 2) Monte Carlo simulation Sekino-Yoneya (’99) for operators corresponding to supergravity modes in 10d SUGRA Hanada-J.N.-Sekino-Yoneya (’09,’11) Actually, agreement extends to M theory regime ! Power-law behavior with the predicted exponent based on Gubser-Klebanov-Polyakov-Witten relation (’98)

14. 1d gauge theory p.b.c. (without loss of generality) 1d SYM with 16 supercharges The region of validity for the SUGRA analysis

15. Series of operators (I) Predicted power law confirmed clearly even beyond the validity region of 10d SUGRA

16. Some details of calculations directly accessible by our Fourier space simulation Gibbs phenomenon ! Actually, Removes the Gibbs phenomenon completely.

17. Series of operators (II) Comparison in the Fourier space : (bad UV behavior) Reliable inverse Fourier tr. seems difficult…

18. Comparison in the Fourier space polynomials of even powers Best fit obtained for

19. Series of operators (III) IR divergent !

20. IR divergent correlation function polynomials of even powers Best fit obtained for finite IR cutoff effects

21. Larger angular momentum Best fit obtained for

23. S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540

24. The action has manifest SO(9,1) symmetry raised and lowered by the metric Hermitian matrices Matrix model proposed as a nonperturbative definition of type IIB superstring theory in 10 dim. Ishibashi-Kawai-Kitazawa-Tsuchiya (’96)

25. 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 Fukuma-Kawai-Kitazawa-Tsuchiya (’98) c.f.) Matrix Theory Banks-Fischler-Shenker-Susskind (’96) Evidence for the conjecture : Aoki-Iso-Kawai-Kitazawa-Tada (’99)

26. Wick rotation Euclidean model SO(10) symmetry opposite sign ! An important feature of the Lorentzian model A conventional approach was: Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01) Partition function becomes finite. SSB of SO(10) J.N.-Okubo-Sugino, arXiv:1108.1293

27. 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 the Euclidean model, but is it really related to the real world ? extended directions shrunken directions

28. connection to the worldsheet theory Unlike the Euclidean model, the path integral is ill-defined ! Nonperturbative dynamics of the Lorentzian model studied, for the first time, in Kim-J.N.-Tsuchiya, arXiv:1108.1540 Introduce IR cutoff in both the temporal and spatial directions They can be removed in the large-N limit. Continuum limit & infinite volume limit

29. Extracting time evolution

30. “critical time” SSB

31. Consider a simpler problem : solution : representation matrices of a compact semi-simple Lie algebra with d generators Maximum is achieved for SU(2) algebra The mechanism of SSB : SO(9) -> SO(3)

32. S.-W.Kim, J.N., A.Tsuchiya, arXiv:1110.4803

33. Lagrange multipliers corresponding to the IR cutoffs Classical equations of motion for the Lorentzian model : We look for a Lie algebraic solution : c.f.) Euclidean model Chatzistavrakidis arXiv:1108.1107 [hep-th]

34. Motivated by Monte Carlo results, we restrict ourselves to and look for solutions with SO(3) symmetry. From the complete list of real Lie algebras with 4 generators the one with SO(3) symmetry is UNIQUE ! Others = 0

35. others = 0 The unitary irreducible representations of can be classified into 2 categories 1) trivial 1d representations 2) infinite-dimensional representations

36. the basis of the functional space Eigenfunctions of the Hamiltonian of a 1d harmonic oscillator

37. SO(3) symmetric solutions Using a direct sum of the non-trivial representations, In what follows,

38. Compatible with the expanding behavior ! size of the space

39. (dimensionless) space-time noncommutativity c.f.) space-time uncertainty principle Yoneya (2000)

40. Speculations time classical solution t cr Monte Carlo simulation SO(9) size of the space space-space noncommutativity present time accelerating expansion space-time noncommutativity Space-space NC disappears for some dynamical reason. symmetry of space SO(3)

42. Black hole singularity Big bang singularity Monte Carlo simulation of supersymmetric gauge theories and matrix models Quantum effects of gravity become crucial. Two kinds of singularity predicted by Einstein’s general relativity Superstring theory Gauge-gravity correspondence “Emergent space” 1d SYM describes the space-time with black hole geometry Lorentzian matrix model Emergence of (3+1)d expanding universe Summary

43. Future directions Extending the study of supersymmetric gauge theory to higher dimensions 1d SYM with mass deformation Large-N equivalence Ishii-Ishiki-Shimasaki-Tsuchiya (2008) superconformal “Holographic inflation” (Skenderis) Connecting the “two ends” in the Lorentzian matrix model  Quantum corrections around the classical solutions  Exploring more general SO(3) symmetric solutions The gauge group and the matter contents, power-law expansion

45. holographic dual of SUSY matrix QM Itzhaki-Maldacena-Sonnenschein-Yankielowicz ’98 near-extremal 0-brane solution in type IIA SUGRA (string frame) dilaton : decoupling limit with fixed (’t Hooft coupling)

46. validity of the SUGRA description : curvature radius (in string units) dilaton at the radius U Black hole thermodynamics internal energy Klebanov-Tseytlin ’96 We check this in strongly coupled gauge theory !

47. corrections to SUGRA action tree-level scattering amplitudes of the massless modes low energy effective action of type IIA superstring theory leading term : type IIA SUGRA action explicit calculations of 2-pt and 3-pt amplitudes 4-pt amplitudes Complete form is yet to be determined, but we can still make a dimensional analysis.

48. Black hole thermodynamics with corrections curvature radius of the dual geometry More careful treatment leads to the same conclusion. (Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602

49. Hanada-J.N.-Takeuchi, PRL 99 (07) 161602 [arXiv:0706.1647] Non-lattice simulation residual gauge symmetry : should be fixed by imposing static diagonal gauge : Note: Gauge symmetry can be fixed non-perturbatively in 1d. c.f.) lattice approach : Catterall-Wiseman, JHEP 0712:104,2007 RHMC algorithm can be used efficiently (Fourier acceleration without extra cost etc.)

50. What is M theory ? hypothetical 11d theory suggested from string dualities low-energy effective theory : 11D SUGRA fundamental d.o.f.: membrane soliton-like objects: M5-brane compactify the theory on a circle 10D type IIA superstring believed to appear in the strong coupling limit of 10D type IIA superstring Witten (’95)

51. Implications on the M theory limit The exponents obtained from10d SUGRA analysis are valid also in the M-theory limit! The exponent agrees with the prediction even at N=3. Surprising aspects of our MC results (2): M theory limit amounts to: