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The origin of space-time as seen from matrix model simulations

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

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The origin of space-time as seen from matrix model simulations

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  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

  2. 1. Introduction

  3. 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, … Quantum gravity Superstring theory is a natural candidate for a unified theory including quantum gravity

  4. The testing ground for superstring theory 2 amazing predictions of Einstein’s general relativity Big bang Black hole 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 • Introduction • Black hole thermodynamics from gauge theory • 3. Direct test of gauge-gravity correspondence • (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. 1. Black hole thermodynamics from gauge theory 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. 1d U(N) SUSY gauge theory Gauge-gravity duality for D0-brane system type IIA superstring Itzhaki-Maldacena-Sonnenschein -Yankielowicz (’98) N D0 branes horizon t black 0-brane solution in type IIA SUGRA near-extremal black hole at finite T In the decoupling limit, the D0 brane system describes the black hole microscopically. SUGRA description : valid

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

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

  11. 3. Direct test of gauge-gravity correspondence M.Hanada, J.N., Y.Sekino, T.Yoneya : Phys.Rev. Lett. 104 (2010) 151601 arXiv: 1108.5153

  12. Prediction from gauge/gravity duality (II) correlation functions in gauge theory generating functional operator-field correspondence gauge gravity Gubser-Klebanov-Polyakov-Witten relation (’98) 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 Sekino-Yoneya (’99) based on Gubser-Klebanov-Polyakov-Witten relation (’98) for operators corresponding to supergravity modes in 10d SUGRA 2) Monte Carlo simulation Hanada-J.N.-Sekino-Yoneya (’09,’11) Power-law behavior with the predicted exponent Actually, agreement extends to M theory regime !

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

  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) Reliable inverse Fourier tr. seems difficult… (bad UV behavior) Comparison in the Fourier space :

  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 finite IR cutoff effects Best fit obtained for

  21. Larger angular momentum Best fit obtained for

  22. Best fit obtained for

  23. 4. (3+1)d expanding universe from matrix model S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540

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

  25. Evidence for the conjecture : • 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) Aoki-Iso-Kawai-Kitazawa-Tada (’99)

  26. An important feature of the Lorentzian model opposite sign ! A conventional approach was: Wick rotation Euclidean model SO(10) symmetry • Partition function becomes finite. Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01) • 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) extended directions shrunken directions 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 ?

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

  29. Extracting time evolution

  30. SSB “critical time”

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

  32. 5. Expanding universe as a classical solution S.-W.Kim, J.N., A.Tsuchiya, arXiv:1110.4803

  33. Classical equations of motion for the Lorentzian model : Lagrange multipliers corresponding to the IR cutoffs 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. The unitary irreducible representations of others = 0 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. Using a direct sum of the non-trivial representations, SO(3) symmetric solutions In what follows,

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

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

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

  41. 6. Summary and discussions

  42. Summary Two kinds of singularity predicted by Einstein’s general relativity • Black hole singularity • Big bang singularity Quantum effects of gravity become crucial. Monte Carlo simulation of supersymmetric gauge theories and matrix models 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

  43. Future directions • Extending the study of supersymmetric gauge theory • to higher dimensions superconformal “Holographic inflation” (Skenderis) Large-N equivalence Ishii-Ishiki-Shimasaki-Tsuchiya (2008) 1d SYM with mass deformation • 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

  44. Backup slides

  45. holographic dual of SUSY matrix QM Itzhaki-Maldacena-Sonnenschein-Yankielowicz ’98 near-extremal0-brane solution in type IIA SUGRA (string frame) dilaton : decoupling limit with fixed (’t Hooft coupling)

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

  47. corrections to SUGRA action low energy effective action of type IIA superstring theory tree-level scattering amplitudes of the massless modes 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. should be fixed by imposing Non-lattice simulation Hanada-J.N.-Takeuchi, PRL 99 (07) 161602 [arXiv:0706.1647] Note: Gauge symmetry can be fixed non-perturbativelyin 1d. • static diagonal gauge : • residual gauge symmetry : RHMC algorithm can be used efficiently (Fourier acceleration without extra cost etc.) c.f.) lattice approach : Catterall-Wiseman, JHEP 0712:104,2007

  50. What is M theory ? Witten (’95) 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

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