Dynamical degrees of freedom for higher genus Riemann surface in (2+1)-dimensional general relativity

TL;DR

This paper develops a homogeneous 2D metric incorporating Teichmüller deformation as dynamical degrees of freedom, providing a concrete formulation for higher genus Riemann surfaces within (2+1)-dimensional general relativity.

Contribution

It introduces a homogeneous standard metric including Teichmüller deformation for higher genus surfaces, advancing the understanding of dynamical degrees of freedom in lower-dimensional gravity.

Findings

Formulation for higher genus Riemann surface using pinching parameter

Homogeneous anisotropic metric satisfying momentum constraints

Inclusion of Teichmüller deformation as dynamical degrees of freedom

Abstract

A homogeneous two-dimensional metric including the degrees of freedom of Teichm\"uller deformation is developed. The Teichm\"uller deformation is incorporated by affine stretching of complex structure. According to Yamada's investigation by pinching parameter, concrete formulation for a higher genus Riemann surface can be realized. We will have a homogeneous standard metric including the dynamical degrees of freedom as Teichm\"uller deformation in a leading order of the pinching parameter, which would be treated as homogeneous anisotropic metric for a double torus universe, which satisfy momentum constraints.