Abstract
In work [1], a surface embedded in flat ℝ 3 is associated to any three hermitian matrices. We study this emergent surface when the matrices are large, by constructing coherent states corresponding to points in the emergent geometry. We find the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes: for example, we examine a round sphere with a non-spherically symmetric Poisson structure. We also give a natural construction for a noncommutative torus embedded in ℝ 3. Finally, we make remarks about area and find matrix equations for minimal area surfaces.
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References
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].
R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].
I. Ellwood, Relating branes and matrices, JHEP 08 (2005) 078 [hep-th/0501086] [INSPIRE].
F. Berezin, General Concept of Quantization, Commun. Math. Phys. 40 (1975) 153.
A.M. Perelomov, Coherent states for arbitrary lie groups, Commun. Math. Phys. 26 (1972) 222 [INSPIRE].
H. Grosse and P. Presnajder, The construction of noncommutative manifolds using coherent states, Lett. Math. Phys. 28 (1993) 239.
W.D. Kirwin, Coherent States in Geometric Quantization, math/0502026.
W.-M. Zhang, D.H. Feng and R. Gilmore, Coherent states: Theory and some Applications, Rev. Mod. Phys. 62 (1990) 867 [INSPIRE].
A.M. Perelomov, Generalized coherent states and their applications, Theoretical and Mathematical Physics, Springer Berlin Heidelberg, Germany (1986) [ISBN:9783540159124].
H. Steinacker, Non-commutative geometry and matrix models, PoS (QGQGS2011) 004 [arXiv:1109.5521] [INSPIRE].
H. Steinacker, Emergent Geometry and Gravity from Matrix Models: an Introduction, Class. Quant. Grav. 27 (2010) 133001 [arXiv:1003.4134] [INSPIRE].
M. Kontsevich, Deformation quantization of Poisson manifolds. 1., Lett. Math. Phys. 66 (2003) 157 [q-alg/9709040] [INSPIRE].
J. Arnlind, M. Bordemann, L. Hofer, J. Hoppe and H. Shimada, Fuzzy Riemann surfaces, JHEP 06 (2009) 047 [hep-th/0602290] [INSPIRE].
J. Arnlind, M. Bordemann, L. Hofer, J. Hoppe and H. Shimada, Noncommutative Riemann Surfaces, arXiv:0711.2588 [INSPIRE].
G. Ishiki, Matrix Geometry and Coherent States, Phys. Rev. D 92 (2015) 046009 [arXiv:1503.01230] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
J. Arnlind and J. Hoppe, The world as quantized minimal surfaces, Phys. Lett. B 723 (2013) 397 [arXiv:1211.1202] [INSPIRE].
A. Schild, Classical Null Strings, Phys. Rev. D 16 (1977) 1722 [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].
A. Fayyazuddin, Y. Makeenko, P. Olesen, D.J. Smith and K. Zarembo, Towards a nonperturbative formulation of IIB superstrings by matrix models, Nucl. Phys. B 499 (1997) 159 [hep-th/9703038] [INSPIRE].
K. Zarembo and Y. Makeenko, An introduction to matrix superstring models, Phys.Usp. 41 (1998) 1.
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].
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de Badyn, M.H., Karczmarek, J.L., Sabella-Garnier, P. et al. Emergent geometry of membranes. J. High Energ. Phys. 2015, 89 (2015). https://doi.org/10.1007/JHEP11(2015)089
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DOI: https://doi.org/10.1007/JHEP11(2015)089