Abstract
Recently we have studied the Lorentzian version of the IIB matrix model as a nonperturbative formulation of superstring theory. By Monte Carlo simulation, we have shown that the notion of time — as well as space — emerges dynamically from this model, and that we can uniquely extract the real-time dynamics, which turned out to be rather surprising: after some “critical time”, the SO(9) rotational symmetry of the ninedimensional space is spontaneously broken down to SO(3) and the three-dimensional space starts to expand rapidly. In this paper, we study the same model based on the classical equations of motion, which are expected to be valid at later times. After providing a general prescription to solve the equations, we examine a class of solutions, which correspond to manifestly commutative space. In particular, we find a solution with an expanding behavior that naturally solves the cosmological constant problem.
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References
H. Liu, G.W. Moore and N. Seiberg, Strings in a time dependent orbifold, JHEP 06 (2002) 045 [hep-th/0204168] [INSPIRE].
H. Liu, G.W. Moore and N. Seiberg, Strings in time dependent orbifolds, JHEP 10 (2002) 031 [hep-th/0206182] [INSPIRE].
A. Lawrence, On the instability of 3D null singularities, JHEP 11 (2002) 019 [hep-th/0205288] [INSPIRE].
G.T. Horowitz and J. Polchinski, Instability of space-like and null orbifold singularities, Phys. Rev. D 66 (2002) 103512 [hep-th/0206228] [INSPIRE].
M. Berkooz, B. Craps, D. Kutasov and G. Rajesh, Comments on cosmological singularities in string theory, JHEP 03 (2003) 031 [hep-th/0212215] [INSPIRE].
T. Banks, W. Fischler, S. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/9612115] [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Matrix string theory, Nucl. Phys. B 500 (1997) 43 [hep-th/9703030] [INSPIRE].
D.Z. Freedman, G.W. Gibbons and M. Schnabl, Matrix cosmology, AIP Conf. Proc. 743 (2005) 286 [hep-th/0411119] [INSPIRE].
B. Craps, S. Sethi and E.P. Verlinde, A matrix big bang, JHEP 10 (2005) 005 [hep-th/0506180] [INSPIRE].
M. Li, A class of cosmological matrix models, Phys. Lett. B 626 (2005) 202 [hep-th/0506260] [INSPIRE].
S.R. Das and J. Michelson, PP wave big bangs: matrix strings and shrinking fuzzy spheres, Phys. Rev. D 72 (2005) 086005 [hep-th/0508068] [INSPIRE].
B. Chen, The time-dependent supersymmetric configurations in M-theory and matrix models, Phys. Lett. B 632 (2006) 393 [hep-th/0508191] [INSPIRE].
J.-H. She, A matrix model for Misner universe, JHEP 01 (2006) 002 [hep-th/0509067] [INSPIRE].
E.J. Martinec, D. Robbins and S. Sethi, Toward the end of time, JHEP 08 (2006) 025 [hep-th/0603104] [INSPIRE].
T. Ishino and N. Ohta, Matrix string description of cosmic singularities in a class of time-dependent solutions, Phys. Lett. B 638 (2006) 105 [hep-th/0603215] [INSPIRE].
T. Matsuo, D. Tomino, W.-Y. Wen and S. Zeze, Quantum gravity equation in large-N Yang-Mills quantum mechanics, JHEP 11 (2008) 088 [arXiv:0807.1186] [INSPIRE].
D. Klammer and H. Steinacker, Cosmological solutions of emergent noncommutative gravity, Phys. Rev. Lett. 102 (2009) 221301 [arXiv:0903.0986] [INSPIRE].
J. Lee and H.S. Yang, Quantum gravity from noncommutative spacetime, SIGMA (2010) [arXiv:1004.0745] [INSPIRE].
H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Space-time structures from IIB matrix model, Prog. Theor. Phys. 99 (1998) 713 [hep-th/9802085] [INSPIRE].
T. Hotta, J. Nishimura and A. Tsuchiya, Dynamical aspects of large-N reduced models, Nucl. Phys. B 545 (1999) 543 [hep-th/9811220] [INSPIRE].
J. Ambjørn, K. Anagnostopoulos, W. Bietenholz, T. Hotta and J. Nishimura, Large-N dynamics of dimensionally reduced 4D SU(N) super Yang-Mills theory, JHEP 07 (2000) 013 [hep-th/0003208] [INSPIRE].
J. Ambjørn, K. Anagnostopoulos, W. Bietenholz, T. Hotta and J. Nishimura, Monte Carlo studies of the IIB matrix model at large N, JHEP 07 (2000) 011 [hep-th/0005147] [INSPIRE].
K. Anagnostopoulos and J. Nishimura, New approach to the complex action problem and its application to a nonperturbative study of superstring theory, Phys. Rev. D 66 (2002) 106008 [hep-th/0108041] [INSPIRE].
K.N. Anagnostopoulos, T. Azuma and J. Nishimura, A practical solution to the sign problem in a matrix model for dynamical compactification, JHEP 10 (2011) 126 [arXiv:1108.1534] [INSPIRE].
J. Nishimura and G. Vernizzi, Spontaneous breakdown of Lorentz invariance in IIB matrix model, JHEP 04 (2000) 015 [hep-th/0003223] [INSPIRE].
J. Nishimura and G. Vernizzi, Brane world from IIB matrices, Phys. Rev. Lett. 85 (2000) 4664 [hep-th/0007022] [INSPIRE].
J. Nishimura and F. Sugino, Dynamical generation of four-dimensional space-time in the IIB matrix model, JHEP 05 (2002) 001 [hep-th/0111102] [INSPIRE].
H. Kawai, S. Kawamoto, T. Kuroki, T. Matsuo and S. Shinohara, Mean field approximation of IIB matrix model and emergence of four-dimensional space-time, Nucl. Phys. B 647 (2002) 153 [hep-th/0204240] [INSPIRE].
T. Aoyama and H. Kawai, Higher order terms of improved mean field approximation for IIB matrix model and emergence of four-dimensional space-time, Prog. Theor. Phys. 116 (2006) 405 [hep-th/0603146] [INSPIRE].
T. Imai, Y. Kitazawa, Y. Takayama and D. Tomino, Quantum corrections on fuzzy sphere, Nucl. Phys. B 665 (2003) 520 [hep-th/0303120] [INSPIRE].
T. Imai, Y. Kitazawa, Y. Takayama and D. Tomino, Effective actions of matrix models on homogeneous spaces, Nucl. Phys. B 679 (2004) 143 [hep-th/0307007] [INSPIRE].
T. Imai and Y. Takayama, Stability of fuzzy S 2 × S 2 geometry in IIB matrix model, Nucl. Phys. B 686 (2004) 248 [hep-th/0312241] [INSPIRE].
J. Nishimura, T. Okubo and F. Sugino, Systematic study of the SO(10) symmetry breaking vacua in the matrix model for type IIB superstrings, JHEP 10 (2011) 135 [arXiv:1108.1293] [INSPIRE].
W. Krauth, H. Nicolai and M. Staudacher, Monte Carlo approach to M-theory, Phys. Lett. B 431 (1998) 31 [hep-th/9803117] [INSPIRE].
P. Austing and J.F. Wheater, Convergent Yang-Mills matrix theories, JHEP 04 (2001) 019 [hep-th/0103159] [INSPIRE].
S.-W. Kim, J. Nishimura and A. Tsuchiya, Expanding (3 + 1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9 + 1)-dimensions, Phys. Rev. Lett. 108 (2012) 011601 [arXiv:1108.1540] [INSPIRE].
S.-W. Kim, J. Nishimura and A. Tsuchiya, Expanding universe as a classical solution in the Lorentzian matrix model for nonperturbative superstring theory, Phys. Rev. D 86 (2012) 027901 [arXiv:1110.4803] [INSPIRE].
M. Fukuma, H. Kawai, Y. Kitazawa and A. Tsuchiya, String field theory from IIB matrix model, Nucl. Phys. B 510 (1998) 158 [hep-th/9705128] [INSPIRE].
J. Ambjørn, J. Jurkiewicz and R. Loll, Reconstructing the universe, Phys. Rev. D 72 (2005) 064014 [hep-th/0505154] [INSPIRE].
H. Kawai and T. Okada, Asymptotically vanishing cosmological constant in the multiverse, Int. J. Mod. Phys. A 26 (2011) 3107 [arXiv:1104.1764] [INSPIRE].
H. Steinacker, Split noncommutativity and compactified brane solutions in matrix models, Prog. Theor. Phys. 126 (2011) 613 [arXiv:1106.6153] [INSPIRE].
A. Chatzistavrakidis, On Lie-algebraic solutions of the type IIB matrix model, Phys. Rev. D 84 (2011) 106010 [arXiv:1108.1107] [INSPIRE].
J. Patera, R. Sharp, P. Winternitz and H. Zassenhaus, Invariants of real low dimension Lie algebras, J. Math. Phys. 17 (1976) 986 [INSPIRE].
J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [INSPIRE].
S. Iso, Y. Kimura, K. Tanaka and K. Wakatsuki, Noncommutative gauge theory on fuzzy sphere from matrix model, Nucl. Phys. B 604 (2001) 121 [hep-th/0101102] [INSPIRE].
J. Nishimura, The origin of space-time as seen from matrix model simulations, Prog. Theor. Exp. Phys. 2012 (2012) 01A101 [arXiv:1205.6870] [INSPIRE].
H. Steinacker, Emergent geometry and gravity from matrix models: an introduction, Class. Quant. Grav. 27 (2010) 133001 [arXiv:1003.4134] [INSPIRE].
M. Hanada, H. Kawai and Y. Kimura, Describing curved spaces by matrices, Prog. Theor. Phys. 114 (2006) 1295 [hep-th/0508211] [INSPIRE].
A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, Intersecting branes and a standard model realization in matrix models, JHEP 09 (2011) 115 [arXiv:1107.0265] [INSPIRE].
H. Aoki, Chiral fermions and the standard model from the matrix model compactified on a torus, Prog. Theor. Phys. 125 (2011) 521 [arXiv:1011.1015] [INSPIRE].
N. Ja. Vilenkin and A.U. Klimyk, Representation of Lie groups and special functions volume 1, Kluwer Academic Publishers, Dordrecht The Netherlands (1991).
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ArXiv ePrint: 1208.0711
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Kim, SW., Nishimura, J. & Tsuchiya, A. Late time behaviors of the expanding universe in the IIB matrix model. J. High Energ. Phys. 2012, 147 (2012). https://doi.org/10.1007/JHEP10(2012)147
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DOI: https://doi.org/10.1007/JHEP10(2012)147