Expanding (3+1)-Dimensional universe from the IIB matrix model

1. Expanding (3+1)-Dimensional universe from the IIB matrix model Asato Tsuchiya (Shizuoka Univ.) SQS ’2013 @Bogoliubov Laboratory, July 29th, 2013

2. References Sang-Woo Kim, Jun Nishimura and A. T. PRL 108 (2012) 011601, arXiv:1108.1540 PRD 86 (2012) 027901, arXiv:1110.4803 JHEP 10 (2012) 147, arXiv:1208.0711 Jun Nishimura and A.T., PTEP 2013, 043B03, arXiv:1208.4910 arXiv: 1305.5547 Hajime Aoki, Jun Nishimura and A. T., in preparation Yuta Ito, Jun Nishimura and A. T., in preparation

3. Present status of superstring theory perturbation theory　＋　D-brane Numerous vacua There are numerous vacua that are theoretically allowed space-time dimensions gauge groups matter contents (generation number) various Use the statistical method or appeal to the anthropic principle Cosmic (initial) singularity Liu-Moore-Seiberg (’02), …… In general, perturbation theory cannot resolve the cosmic singularity Non-perturbative effects are important at the beginning of the universe

4. Matrix models There is a possibility that one can actually determine the true vacuum uniquely and resolve the cosmic singularity if one uses a nonperturbative formulation that incorporates full nonperturbative effects. ~ lattice QCD for QCD Proposals IIB matrix model Ishibashi-Kawai-Kitazawa-A.T. (’96) 0D 10D U(N)SYM Matrix theory Banks-Fischler-Shenker-Susskind (‘96) 1D Matrix string theory Dijkgraaf-Verlinde-Verlinde (’97) 2D We study the IIB matrix model. “Lorentzian” is a key to this project.

5. Plan of the present talk • Introduction • Review of the IIB matrix model • Defining Lorentzian version • SSB of SO(9) symmetry to SO(3) • Realizing the Standard Model particles • Summary and outlook

6. IIB matrix model

7. IIB matrix model Ishibashi-Kawai-Kitazawa-A.T. (’96) Hermitian matrices : 10D Lorentz vector : 10D Majorana-Weyl spinor Large- limit is taken Space-time does not exist a priori, but is generated dynamically Time is given a priori in Matrix theory and matrix string theory Manifest SO(9,1) symmetry, manifest 10D N=2 SUSY covariant formulation Matrix theory and matrix string theory are light-cone formulation

8. Correspondence with world-sheet action Green-Schwarz action of Schild-type for type IIB superstrig with κ symmetry fixed matrix regularization IIB matrix model 2nd quantized

9. 10D N=2 SUSY Corresponding to 10DN=2SUSYpossessed by Green-Schwarz action translation of eigenvalues dimensional reduction of 10D N=1 SUSY 10D N=2 SUSY eigenvalues of are coordinates suggests that the model includes gravity

10. Interaction between D-branes graviton scalar

11. Light-cone string field theory Fukuma-Kawai-Kitazawa-A.T. (’97) Schwinger-Dyson equation for on the light front light-cone string field theory for type IIB superstring This implies that IIB matrix modelreproduces perturbation theory of type IIB superstring

12. Euclidean model Lorentzian model looks quite unstable opposite sign！ Euclideanization Wickrotation Euclidean model manifest SO(10) symmetry : positive definite Euclidean model is well-defined without cutoffs Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01) People have been studying the Euclidean model

13. Space-time in Euclidean model configurationsdiagonalizable simultaneously are favored SSB of SO(10) to SO(4) 10 hermitian matrices 4Ｄ 10Ｄ low energy effective theory for ~ branched polymer Aoki-Iso-Kawai-Kitazawa-Tada (’99) If eigenvalues are distributed on 4D hyperplane, SSB of SO(10) to SO(4)occurs ~ dynamical generation of 4D space-time But not confirmed so far

14. Defining Lorentzian version Kim-Nishimura-A. T. (’11)

15. Why Lorentzian version? • see time evolution of universe ~ need to study real time dynamics • Wick rotation in gravitational theory seems nontrivial, in contrast to field theory on flat space-time ex.) causal dynamical triangulation （CDT) Ambjorn-Jurkiewicz-Loll (’05) Colemanmechanism in space-time with Lorenztian signature Kawai-Okada (’11) • Recent resultsfor the Euclidean model in Gaussian expansion methodNishimura-Okubo-Sugino (’11) suggest dynamical generation of 3-dimensional space-time Here we study Lorentzian version of the IIB matrix model

16. Regularization and large-N limit • How to define partition function natural from a viewpoint of the Wick rotation of worldsheet • By introducing cutoffs, we make time and space directions finite • It turns out that these cutoffs can be removed in the large-N limit • ~quite nontrivial dynamical property • Lorentzian version is well-defined • we expect the effect of the explicit breaking of SO(9,1) and SUSY • due to the cutoffs to vanish in the large-N limit need to check

17. SSB of SO(9) to SO(3) Kim-Nishimura-A. T. (’11)

18. Emergence of concept of ``time evolution” average these values are determined dynamically small We observe band-diagonal structure represents space structure at fixed time t small

19. Determination of block size We take

20. SSB of SO(9) symmetry SSB symmetric under we only show “critical time”

21. Exponential expansion inflation

22. From exponential to linear 6d bosonic modelomit fermions inflation radiation dominated universe Big Bang?

23. Realizing Standard Model particles Nishimura-A. T. (’12) Aoki-Nishimura-A. T., work in progress Cf.) Chatzistavrakidis-Steinnacker-Zoupanos (’11)

24. Late time behaviors • It is likely that we see the beginning of universe (inflation) • in the present Monte Carlo simulation • we need larger N to see late times • At later times, ``well-known” universe should emerge • Do inflation and big bang occur? not phenomenological description by ``inflaton” but first principle description by superstring theory • present accelerating expansion (dark energy) understanding of cosmological constant problem • The Standard Model should appear • we can expect that at late times classical approximation is good • and fluctuation around the background is small because the action • is large due to expansion

25. Chiral fermions is Majorana-Weyl in 10d Dirac equation in 10d massless modes chiral fermions in 4d

26. Background (3+1)d Lorentz symmetric background matrix analog of warp factor constructive definition # of zero modes with each chirality is the same and are different

27. Warped geometry and chiralfermion • direct product vector-like fermions no-go theorem • warped geometry many solutions # of variables < # of equations no solution generically we obtain left-handed chiral fermion in 4d

28. Example intersecting branes fuzzy spheres 6 ~ D5-brane 5 4 ~ D7-brane 4 5 6 9 7 8

29. Solving 6d Dirac equation R L charge conjugation ~ identified we solve zero modes ( ) appear at intersection points ~ L and R

30. Wave functions and warp factor Left-handed 1st excited state zero modes 6 4, 5 Right-handed 4, 5 we obtain one generation of left-handed fermion 6

31. Three generations we squash L R L R L R we obtain three generations of left-handed fermions wave functions are different among generations so are Yukawa coupling

32. Standard Model fermions and right-handed neutrino Cf.) Chatzistavrakidis-Steinnacker-Zoupanos (’11)

33. Summary and outlook

34. Summary • We studied Lorentzian version of the IIB matrix model • We introduced the infrared cutoffs and found that they can be • removed in the large-N limit • The model thus obtained is well-defined and has no parameters • except one scale parameter This property is expected in nonperturbative string theory • the concept of ``time evolution” emerges when is made diagonal, have band-diagonal structure • After a critical time, SO(9) symmetry of 9 dimension is • spontaneously broken down to SO(3) and 3 out of 9 dimensions • start to expand rapidly ~ can be interpreted as birth of universe

35. Summary (cont’d) • We confirm that the expansion is exponential • ~ beginning of inflation • We observe transition from exponential expansion to linear one • in toy model • We found that matrix analog of warp factor must be introduced to • realize chiral fermions, and gave an example of background • yielding the chiral fermion • We gave an example of background where three generations of the • Standard Model particles are realized • There is a possibility that Standard Model can be derived uniquely • at low energy structure of extra dimensions and warp factor are • determined dynamically ~ to substantiate string theory • We found a solution which represents expanding (3+1)-dimensional • universe and resolves cosmological constant problem

36. Outlook • calculation of Yukawa couplings : mass hierarchy , CKM matrix and • MNS matrix • background at late time : e-foldings, Big Bang, Standard Model • more efficient Monte Carlo simulation ~ parallelization • It is important to study the classical solutions and look for a solution • connected smoothly to the simulation Kim-Nishimura-A. T. (’12) • use the idea of renormalization group to reach the late time • ~ work for a toy model

37. Outlook (cont’d) • Does exponential expansion occur? • Does big bang occur after that? • it should be seen as (the second) phase transition • Does it occur at the same time as commutative space-time appears • How density fluctuation can be measured to compare with CMB • Can lagrangian of GUT be read off from fluctuation around the • classical solution? • Does standard model appear at low energy? • We expect to understand in a unified way various problems in • particle physics and cosmology such as • mechanism of inflation, cosmological constant problem, • hierarchy problem, dark matter, dark energy, baryogenesis

38. String duality dualities in superstring theory M IIA Het E8 x E8 five superstring theories and M theory are different descriptions of one theory Het SO(32) IIB I One can start everywhere with a formulation which enables one to treat strong coupling dynamics

39. Lorentzian version How to define partition function natural from a viewpoint of correspondence with worldsheet theory Wickrotation for worldsheet

40. Regularization and large-N limit unlike the case of the Euclidean model, the Lorentzian model is ill-defined as it stands • By introducing cutoffs, we make time and space directions finite without loss of generality we put • It turns out that these cutoffs can be removed in the large-N limit • ~quite nontrivial dynamical property • SO(9,1) symmetry and SUSY are explicitly broken by the cutoffs • we expect the effect of the explicit breaking to vanish • in the large-N limit need to check

41. Avoiding sign problem （２） （１） can give rise to sign problem （１）　Pfaffian coming from integral over fermion is dominant in the large-N limit in the Euclidean model

42. Avoiding sign problem (cont’d) homogeneous in Also problem in field theory on Minkowski space （analysis of real time dynamics is a notorious problem） （２） How to treat first perform

43. Time evolution of space size peak at starts to grow for Symmetric under We only show

44. Late time behaviors

45. Seeing later times • It is likely that we see the beginning of universe in the present • Monte Carlo simulation • At later times, ``well-known” universe should emerge • Do inflation and big bang occur? not phenomenological description by ``inflaton” but first principle description by superstring theory can be tested by comparison with CMB • How commutative space-time appears • present accelerating expansion (dark energy) understanding of cosmological constant problem • What massless fields appear on it • prediction for fate of universe big crunch or big rip

46. Classical approximation as a complementary approach • we expect classical approximation is good after late times • because action becomes large due to expansion • there are infinitely many solutions • if we reach the time by Monte Carlo simulation and uniquely • pick up a dominant classical solution connected smoothly to • the result in the simulation, we can study late time behaviors

47. Example of solution • variation function • EOM • SL(2,R) solution (3+1)-dimensional space-time～R×S3

48. Cosmological implication of SL(2,R) solution considered to give late time behaviors in the matrix model identify t0 with the present time present accelerating expansion cosmological constant ~ solve cosmological constant problem cosmological constant term vanishes in the future

49. Chiral fermions cont’d nonzero eigenvalues are paired finite-N matrices ~space of with each chirality has the same dimensions ~number of zero modes with each chirality is the same

50. Commutative space-time and local fields • results in Monte Carlo simulation are smaller than • there are classical solutions representing (3+1)-dimensional • commutative space-time • we assume a classical solution representing (3+1)-dimensional • commutative space-time is dominant at sufficiently late times are close to diagonal space-time points in 3+1 dimensions are small for with