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3D out of 9D ---the birth of our Universe from the superstring theory on computer

3D out of 9D ---the birth of our Universe from the superstring theory on computer. string seminar at UBC Vancouver, Canada Nov. 5, 2012. Jun Nishimura (KEK & SOKENDAI). Plan of the talk. Introduction Problems in the Euclidean model Lorentzian matrix model The birth of our Universe

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3D out of 9D ---the birth of our Universe from the superstring theory on computer

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  1. 3D out of 9D ---the birth of our Universe from the superstring theory on computer string seminar at UBC Vancouver, Canada Nov. 5, 2012 Jun Nishimura (KEK & SOKENDAI)

  2. Plan of the talk • Introduction • Problems in the Euclidean model • Lorentzian matrix model • The birth of our Universe • The time-evolution at later times • Towards the derivation of a grand unified theory • Summary and future prospects

  3. collaborators • Asato Tsuchiya(Shizuoka Univ.) • Sang-Woo Kim (Osaka Univ.) • Konstantinos Anagnostopoulos (National Technical University, Athens) • Yuta Ito (SOKENDAI) • Yuki Koizuka (SOKENDAI)

  4. 1. Introduction

  5. 1. Introduction Nonperturbative formulation is important in QFT e.g.) quark confinement in QCD Lattice gauge theory (Wilson, 1974) Area law Strong coupling expansion, MC simulation A property which can never be explained by perturbation theory!

  6. This might also be the case in superstring theory. Since 1980s, it has been thought that superstring theory has infinitely many perturbative vacua … Space-time dimensionality Gauge symmetry Matter (#generations) depend on the vacuum “Landscape” (= “What’s wrong with it?”) Our Universe may be just one of the infinitely many possible vacua. Therefore, the goal of superstring theory is to construct such a vacuum explicitly and to study its property. However, There is also a possibility that the vacuum is uniquely determined if the theory is formulated nonperturbatively.

  7. Type IIB matrix model (Ishibashi, Kawai, Kitazawa, Tsuchiya, 1996) Nonperturbative formulation of superstring theory based on type IIB superstring theory in 10d. M • The connection to perturbative formulations can be seen manifestly • by considering 10d type IIB superstring theory in 10d. IIA Het E8 x E8 Het SO(32) IIB worldsheet action, light-cone string field Hamiltonian, etc. I • A natural extension of the “one-matrix model”, which is established as • a nonperturbative formulation of non-critical strings regarding Feynman diagrams in matrix models as string worldsheets. • It is expected to be a nonperturbative formulation of • the unique underlying theory of string dualities (Other types of superstring theory can be represented as perturbative vacua of type IIB matrix model)

  8. Hermitian matrices Type IIB matrix model SO(9,1) symmetry Lorentzian metric is used to raise and lower indices Wick rotation Euclidean matrix modelSO(10) symmetry

  9. Lorenzian v.s. Euclidean There was a reason why no one dared to study the Lorentzian model for 15 years! opposite sign! extremely unstable system. Once one Euclideanizes it, positive definite! flat direction ()is lifted up due to quantum effects. Aoki-Iso-Kawai-Kitazawa-Tada (’99) Euclidean model is well difined without the need for cutoffs. Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01)

  10. 2. Problems in the Euclidean model

  11. Previous works in the Euclidean matrix model A model with SO(10) rotational symmetry instead of SO(9,1) Lorentz symmetry Dynamical generation of 4d space-time ? SSB of SO(10) rotational symmetry • perturbative expansion around diagonal configurations, • branched-polymer picture • Aoki-Iso-Kawai-Kitazawa-Tada(1999) • The effect of complex phase of the fermion determinant (Pfaffian) • J.N.-Vernizzi (2000) • Monte Carlo simulation • Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(2000) • Anagnostopoulos-J.N.(2002) • Gaussian expansion method • J.N.-Sugino (2002)、Kawai-Kawamoto-Kuroki-Matsuo-Shinohara(2002) • fuzzy • Imai-Kitazawa-Takayama-Tomino(2003)

  12. Most recent results based on the Gaussian expansion method J.N.-Okubo-Sugino, JHEP1110(2011)135, arXiv:1108.1293 extended direction shrunken direction d=3 gives the minimum free energy Space-time has finite extent in all directions ＳＳＢ ＳＯ（１０）　　　　　　ＳＯ（３） This is an interesting dynamical property of the Euclideanized type IIB matrix model. However, its physical meaning is unclear…

  13. After all, the problem was in the Euclideanization ? • In QFT, it can be fully justified as an analytic continuation. • 　　　　（That’s why we can use lattice gauge theory.) • On the other hand, it is subtle in gravitating theory. • (although it might be OK at the classical level…) • Quantum gravity based on dynamical triangulation (Ambjorn et al. 2005） • (Problems with Euclidean gravity may be overcome in Lorentzian gravity.) • Coleman’s worm hole scenario for the cosmological constant problem • (A physical interpretation is possible only by considering the Lorentzian • version instead of the original Euclidean version.) Okada-Kawai (2011) • Euclidean theory is useless for studying the real time dynamics • such as the birth of our Universe.

  14. 3. Lorentzian matrix model

  15. Monte Carlo simulation of Lorentzian type IIB matrix model Kim-J.N.-TsuchiyaPRL 108 (2012) 011601[arXiv:1108.1540] How to define the partition function This seems to be natural from the connection to the worldsheet theory. (The worldsheet coordinates should also be Wick-rotated.)

  16. Regularization and the large-N limit Unlike the Euclidean model, the Lorentzian model is NOT well defined as it is. • The extent in temporal and spatial directions should be made finite. • (by introducing cutoffs) In what follows, we set without loss of generality. • It turned out that these two cutoffs can be removed in the large-N limit. • (highly nontrivial dynamical property) • Both SO(9,1) symmetry and supersymmetry are broken explicitly by the cutoffs. • The effect of this explicit breaking is expected to disappear • in the large-N limit.（needs to be verified.)

  17. Sign problem can be avoided ! (2) (1) The two source of the problem. (1)Pfaffian coming from integrating out fermions The configurations with positive Pfaffian become dominant at large N. In Euclidean model, This complex phase induces the SSB of SO(10) symmetry. J.N.-Vernizzi (’00), Anagnostopoulos-J.N.(’02)

  18. Sign problem can be avoided ! (2) What shall we do with The same problem occurs in QFT in Minkowski space (Studying real-time dynamics in QFT is a notoriously difficult problem.) First, do homogenous in

  19. 4. The birth of our Universe

  20. Emeregence of the notion of “time-evolution” mean value small band-diagonal structure represents the state at the time t small

  21. The emergence of “time” Supersymmetry plays a crucial role! Calculate the effective action for at one loop. contributes contributes Contribution from van der Monde determinant Altogether, Zero, in a supersymmetric model ! Attractive force between the eigenvalues in the bosonic model, cancelled in supersymmetric models.

  22. The time-evolution of the extent of space symmetric under We only show the region

  23. SSB of SO(9) rotational symmetry SSB “critical time”

  24. 5. The time-evolution at later times S.-W. Kim, J. Nishimura and A.Tsuchiya, Phys. Rev. D86 (2012) 027901 [arXiv:1110.4803] S.-W. Kim, J. Nishimura and A.Tsuchiya, JHEP 10 (2012) 147 [arXiv:1208.0711]

  25. What can we expect by studying the time-evolution at later times • What is seen by Monte Carlo simulation so far is: • the birth of our Universe • What has been thought to be the most difficult • from the bottom-up point of view, can be studied first. • This is a typical situation in a top-down approach ! • We need to study the time-evolution at later times • in order to see the Universe as we know it now! • Does inflation and the Big Bangoccurs ? • (First-principles description based on superstring theory, instead of just a • phenomenological description using “inflaton”; comparison with CMB etc.. • How does the commutative space-timeappear ? • What kind of massless fields appear on it ? • accelerated expansion of the present Universe (dark energy), • understanding the cosmological constant problem • prediction for the end of the Universe(Big Crunch or Big Rip or...)

  26. Classical approximation as a complementary approach The cosmic expansion makes each term in the action larger at later times classical approximation becomes valid Once we can reach that point by Monte Carlo simulation, all that remains to be done is to just uniquely select the classical solution that is smoothly connected to the Monte Carlo results. Rem.) There are infinitely many classical solutions. (similarly to the Landscape.) There is a simple solution representing a (3+1)D expanding universe, which naturally solves the cosmological constant problem. By studying the fluctuation around the solution, one should be able to derive GUT.

  27. General prescription • variational function • classical equations of motion • commutation relations Eq. of motion & Jacobi identity Lie algebra Unitary representation classical solution

  28. Ansatz extra dimension is small (compared with Planck scale) commutative space

  29. Simplification Lie algebra e.g.)

  30. d=1 case SO(9) rotation Take a direct sum distributed on a unit S3 (3+1)D space-time R×S3 A complete classification of d=1 solutions has been done. Below we only discuss a physically interesting solution.

  31. SL(2,R) solution • SL(2,R) solution • realization of the SL(2,R) algebra on

  32. Space-time structure in SL(2,R) solution • primary unitary series representation tri-diagonal • 3K×3K diagonal block Space-time noncommutativity disappears in the continuum limit.

  33. Cosmological implication of SL(2,R) solution • the extent of space cont. lim. • Hubble constant and the w parameter radiation dominant matter dominant cosmological constant

  34. Cosmological implication of SL(2,R) solution (cont’d) This part is considered to give the late-time behavior of the matrix model t0 is identified with the present time. present accelerated expansion cosmological const. a solution to the cosmological constant problem Cosmological constant disappears in the future.

  35. 6. Towards the derivation of a grand unified theory J. Nishimura and A.T., arXiv:1208.4910

  36. Commutative space-time and local fields Monte Carlo results are much smaller than Assume that commutative space-timeappears at sufficiently late times. （It can be realized as classical solutions.) is close to diagonal A space-time point in (3+1)D space-time These are considered to be massive modes. Massless modes can be naturally identified as local fields. c.f.) In arXiv:1208.4910, we considered the NG modes associated with the SSB of Poincare symmetry and SUSY as an example.

  37. cf.) Iso-Kawai (’99) Gauge symmetry Suppose we have a classical solution is also a solution By considering fluctuations around it, we obtain local SU(k) symmetry naturally. c.f.) This is analogous to the emergence of SU(k) gauge theory as an effective theory of k coinciding D-branes.

  38. c.f.) H.Aoki PTP 125 (2011) 521 Chatzistavrakidis-Steinacker-Zoupanos JHEP 09 (2011) 115 GUT minimum 3 2 1 1 1 : bi-fundamental rep. of : vector-like partners hypercharge can be assigned consistently is an appropriate linear combination of

  39. GUT (cont’d) • Given a classical solution, one can read off the details of • the local field theory from the fluctuation around it. • Is there a classical solution which gives rise to chiral fermions ? • Does SUSY remain ? • (This question can be answered for any given classical solution.) • If yes, the hierarchy problem is solved. • If no, all the scalar fields acquire mass of the order of GUT scale • due to radiative corrections. • SM Higgs should be considered as composite. c.f.) H.Aoki PTP 125 (2011) 521 Chatzistavrakidis-Steinacker-Zoupanos JHEP 09 (2011) 115 The structure of the extra dimension is crucial.

  40. 7. Summary and future prospects

  41. Summary Type IIB matrix model (1996) A nonperturbative formulation of superstring theory based on type IIB theory in 10d. (15 years since its proposal) The problems with the Euclideanized model have become clear. Lorentzian model : untouched so far because of its instability Monte Carlo simulation has revealed its surprising properties : • a well-defined theory can be obtained by introducing cutoffs • and removing them in the large-N limit • (includes no parameter except for one scale parameter.) • emergence of the notion of “time evolution” When we diagonalize , has band-diagonal structure In order for the eigenvalue distribution of to extend to infinity, SUSY is crucial. • After some “critical time”, the space undergoes the SSB of SO(9), • and only 3 directions start to expand. Can be interpreted as representing the birth of our Universe.

  42. Seiberg’s rapporteur talk (2005) at the 23rd Solvay Conference in Physics “Emergent Spacetime” hep-th/0601234 Understanding how time emerges will undoubtedly shed new light on some of the most important questions in theoretical physics including the origin of the Universe. Indeed in the Lorentzian matrix model, not only space but also time emerges, and the origin of the Universe seems to be clarified.

  43. The significance of the unique determination of the space-time dimensionality It strongly suggests that superstring theory has a unique nonperturbative vacuum. By studying the time-evolution further, one should be able to see the emergence of commutative space-time and massless fields propagating on it. It is conceivable that the SM can be derived uniquely. This amounts to “proving” the superstring theory. It is sufficient to identify the classical configuration which dominates at late timesby studying the time-evolution at sufficiently late times. Independently of this, it is important to study classical solutions and to study the fluctuations around them. Does chiral fermions appear ? Is SUSY preserved ? The key lies in the structure in the extra dimensions.

  44. Future prospects • Does exponential expansion occurs ? How will it be later ? • (Classical solutions suggest that it stops. • Can we see it by Monte Carlo simulation ?) • Does the Big Bang occur after inflation ? • (It should be seen as “the second phase transition”.) • Does the transition to commutative space-time (classical solution) • occur at the same time ? • Can we calculate the density fluctuationto be compared with CMB ? • The value of k of SU(k)gauge symmetryis determined by the solution. • Can we read off the Lagrangian of the SU(k) GUT from the fluctuation ? • Does Standard Model appear at low energy ? Various fundamental questions in particle physics and cosmology : The mechanism of inflation, the cosmological constant problem, the hierarchy problem, dark matter, dark energy, baryogenesis, etc.. It should be possible to understand all these problems in a unified manner.

  45. Appendix: evidence for exponential expansion ＶＤＭ model It looks consistent with exponential expansion.

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