Relativistic Kinematics of Two-Parametric Riemann Surface in Genus Two

It is considered a model of compact Riemann surface in genus two, represented geometrically by two-parametric hyperbolic octagon with an order four automorphism and described algebraically by the corresponding Fuchsian group. Introducing the Fenchel--Nielsen variables, we compute the Weil--Petersson (WP) symplectic two-form for parameter space and analyze the closed isoperimetric orbits of octagons. WP-Area in parameter space and the canonical action--angle variables for the orbits are found. Exploiting the ideas from the loop quantum gravity, we generate relativistic kinematics by the Lorentz boost and quantize WP-area. We treat the evolution in terms of global variables within the "big bounce" concept.