Area Quantization of the Parameter Space of Riemann Surfaces in Genus Two

  • A. V. Nazarenko Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
Keywords: Riemann surfaces in genus two, geometrodynamics, area quantization

Abstract

We consider a model of compact Riemann surfaces in genus two represented geometrically by two-parametric hyperbolic octagons with an order п/2 automorphism. We compute the generators of the Fuchsian group and give a real-analytic description of the corresponding Teichm¨uller space parametrized by the Fenchel–Nielsen variables in terms of geometric data. We state the structure of the parameter space by computing the Weil–Petersson (WP) symplectic
two-form and analyzing the isoperimetric orbits. Combining these results, the WP area in the parameter space and the canonical action–angle variables for the orbits are found. Using the ideas from the loop quantum gravity, we apply our formalism to the description of the classical geometrodynamics of Riemann surfaces and the WP area quantization. The results of the paper may be interesting due to their applications to the quantum geometry, chaotic systems, and low-dimensional gravity.

References

E. D'Hocker and D.H. Phong, Rev. Mod. Phys. 60, 917 (1988).

https://doi.org/10.1103/RevModPhys.60.917

N.E. Hurt, Geometric Quantization in Action: Applications of Harmonic Analysis in Quantum Statistical Mechanics and Quantum Field Theory (Reidel, Dordrecht, 1983).

https://doi.org/10.1007/978-94-009-6963-6

A.V. Nazarenko, Int. J. Mod. Phys. B25, 3415 (2011); e-print arXiv: math-ph/1112.2278 (2011).

M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).

https://doi.org/10.1007/978-1-4612-0983-6

R. Aurich, M. Sieber, and F. Steiner, Phys. Rev. Lett. 61, 483 (1988).

https://doi.org/10.1103/PhysRevLett.61.483

H. Ninnemann, Int. J. Mod. Phys. B9, 1647 (1995).

https://doi.org/10.1142/S0217979295000719

J.E. Nelson and T. Regge, Commun. Math. Phys. 141, 211 (1991).

https://doi.org/10.1007/BF02100010

R. Loll, Class. Quant. Grav. 12, 1655 (1995); e-print arXiv: gr-qc/9408007 (2004).

A. Aigon-Dupuy, P. Buser, M. Cibils, A.F. Kunzle, F. Steiner, J. Math. Phys. 46, 033513 (2005).

https://doi.org/10.1063/1.1850177

S.A. Wolpert, Amer. J. Math. 107, 969 (1985).

https://doi.org/10.2307/2374363

R. Silhol, Comment. Math. Helv. 82, 413 (2007).

https://doi.org/10.4171/CMH/97

R.M. Kashaev, e-print arXiv: math/0008148 (2000).

E.R. Livine and M. Martin-Benito, Phys. Rev. D85, 124052 (2012); e-print arXiv: gr-qc/1204.0539 (2012).

B.A. Dubrovin, S.P Novikov, A.T. Fomenko, Modern Geometry. Methods and Applications (Springer, Berlin, 1984-1990).

P. Buser and R. Silhol, Geometriae Dedicata 115, 121 (2005).

https://doi.org/10.1007/s10711-005-6908-z

P. Buser, Geometry and Spectra of Compact Riemann Surfaces (Birkh¨auser, Basel, 1992).

S. Wolpert, Comment. Math. Helv. 56, 132 (1981).

https://doi.org/10.1007/BF02566203

Y. Imayoshi and M. Taniguchi, An Introduction to Teichm¨uller Space (Springer, Tokyo, 1992).

https://doi.org/10.1007/978-4-431-68174-8

S.A. Wolpert, Ann. of Math. 115, 501 (1982).

https://doi.org/10.2307/2007011

S.A.Wolpert, e-print arXiv: math.DG/0801.0175v1 (2008).

A.V. Nazarenko, e-print arXiv: math-ph/13015446 (2013).

Published
2018-10-11
How to Cite
Nazarenko, A. (2018). Area Quantization of the Parameter Space of Riemann Surfaces in Genus Two. Ukrainian Journal of Physics, 58(11), 1055. https://doi.org/10.15407/ujpe58.11.1055
Section
Archive