Abstract
We consider the commutative limit of matrix geometry described by a large-N sequence of some Hermitian matrices. Under some assumptions, we show that the commutative geometry possesses a Kähler structure. We find an explicit relation between the Kähler structure and the matrix configurations which define the matrix geometry. We also discuss a relation between the matrix configurations and those obtained from the geometric quantization.
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T. Banks, W. Fischler, S.H. 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].
B. de Wit, J. Hoppe and H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].
J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [INSPIRE].
N.M.J. Woodhouse, Geometric quantization, Clarendon Press, Oxford U.K. (1992).
E. Lerman, Geometric quantization; a crash course, arXiv:1206.2334.
D. Berenstein and E. Dzienkowski, Matrix embeddings on flat R 3 and the geometry of membranes, Phys. Rev. D 86 (2012) 086001 [arXiv:1204.2788] [INSPIRE].
L. Schneiderbauer and H.C. Steinacker, Measuring finite quantum geometries via quasi-coherent states, J. Phys. A 49 (2016) 285301 [arXiv:1601.08007] [INSPIRE].
T. Asakawa, S. Sugimoto and S. Terashima, D-branes, matrix theory and K homology, JHEP 03 (2002) 034 [hep-th/0108085] [INSPIRE].
S. Terashima, Noncommutativity and tachyon condensation, JHEP 10 (2005) 043 [hep-th/0505184] [INSPIRE].
G. Ishiki, Matrix geometry and coherent states, Phys. Rev. D 92 (2015) 046009 [arXiv:1503.01230] [INSPIRE].
J. Arnlind, J. Hoppe and G. Huisken, Multi-linear formulation of differential geometry and matrix regularizations, J. Diff. Geom. 91 (2012) 1 [arXiv:1009.4779] [INSPIRE].
G. Alexanian, A.P. Balachandran, G. Immirzi and B. Ydri, Fuzzy CP 2, J. Geom. Phys. 42 (2002) 28 [hep-th/0103023] [INSPIRE].
A. Connes, M.R. Douglas and A.S. Schwarz, Noncommutative geometry and matrix theory: compactification on tori, JHEP 02 (1998) 003 [hep-th/9711162] [INSPIRE].
J. Arnlind, M. Bordemann, L. Hofer, J. Hoppe and H. Shimada, Fuzzy Riemann surfaces, JHEP 06 (2009) 047 [hep-th/0602290] [INSPIRE].
J. Castelino, S. Lee and W. Taylor, Longitudinal five-branes as four spheres in matrix theory, Nucl. Phys. B 526 (1998) 334 [hep-th/9712105] [INSPIRE].
H. Grosse and P. Prešnajder, The construction on noncommutative manifolds using coherent states, Lett. Math. Phys. 28 (1993) 239 [INSPIRE].
H. Grosse and P. Prešnajder, The Dirac operator on the fuzzy sphere, Lett. Math. Phys. 33 (1995) 171 [INSPIRE].
A.B. Hammou, M. Lagraa and M.M. Sheikh-Jabbari, Coherent state induced star product on R 3 λ and the fuzzy sphere, Phys. Rev. D 66 (2002) 025025 [hep-th/0110291] [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].
M.H. de Badyn, J.L. Karczmarek, P. Sabella-Garnier and K. H.-C. Yeh, Emergent geometry of membranes, JHEP 11 (2015) 089 [arXiv:1506.02035] [INSPIRE].
J.L. Karczmarek and K. H.-C. Yeh, Noncommutative spaces and matrix embeddings on flat R2n+1, JHEP 11 (2015) 146 [arXiv:1506.07188] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].
G. Ishiki, S. Shimasaki, Y. Takayama and A. Tsuchiya, Embedding of theories with SU(2|4) symmetry into the plane wave matrix model, JHEP 11 (2006) 089 [hep-th/0610038] [INSPIRE].
J.M. Maldacena, M.M. Sheikh-Jabbari and M. Van Raamsdonk, Transverse five-branes in matrix theory, JHEP 01 (2003) 038 [hep-th/0211139] [INSPIRE].
T. Ishii, G. Ishiki, S. Shimasaki and A. Tsuchiya, N = 4 super Yang-Mills from the plane wave matrix model, Phys. Rev. D 78 (2008) 106001 [arXiv:0807.2352] [INSPIRE].
S. Catterall and T. Wiseman, Towards lattice simulation of the gauge theory duals to black holes and hot strings, JHEP 12 (2007) 104 [arXiv:0706.3518] [INSPIRE].
K.N. Anagnostopoulos, M. Hanada, J. Nishimura and S. Takeuchi, Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature, Phys. Rev. Lett. 100 (2008) 021601 [arXiv:0707.4454] [INSPIRE].
M. Hanada, H. Kawai and Y. Kimura, Describing curved spaces by matrices, Prog. Theor. Phys. 114 (2006) 1295 [hep-th/0508211] [INSPIRE].
H. Steinacker, Emergent gravity from noncommutative gauge theory, JHEP 12 (2007) 049 [arXiv:0708.2426] [INSPIRE].
H. Steinacker, The curvature of branes, currents and gravity in matrix models, JHEP 01 (2013) 112 [arXiv:1210.8364] [INSPIRE].
Y. Ito, J. Nishimura and A. Tsuchiya, Power-law expansion of the universe from the bosonic Lorentzian type IIB matrix model, JHEP 11 (2015) 070 [arXiv:1506.04795] [INSPIRE].
M. Hanada, Y. Hyakutake, G. Ishiki and J. Nishimura, Numerical tests of the gauge/gravity duality conjecture for D0-branes at finite temperature and finite N , arXiv:1603.00538 [INSPIRE].
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ArXiv ePrint: 1603.09146
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Ishiki, G., Matsumoto, T. & Muraki, H. Kähler structure in the commutative limit of matrix geometry. J. High Energ. Phys. 2016, 42 (2016). https://doi.org/10.1007/JHEP08(2016)042
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DOI: https://doi.org/10.1007/JHEP08(2016)042