Orbifold Riemann surfaces and geodesic algebras

TL;DR

This paper explores the Teichmüller theory of orbifold Riemann surfaces with order-two points, extending fat graph techniques to include quantization, mapping-class group transformations, and geodesic algebra structures.

Contribution

It extends existing methods to orbifold surfaces, detailing classical and quantum geodesic algebras and braid group relations for specific algebra types.

Findings

Classical and quantum braid group relations for A_n and D_n algebras.

Description of central elements in Poisson and quantum algebras.

Extension of fat graph quantization techniques to orbifold points.

Abstract

We study the Teichm\"uller theory of Riemann surfaces with orbifold points of order two using the fat graph technique. The previously developed technique of quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebras of geodesic functions is applicable to the surfaces with orbifold points. We describe classical and quantum braid group relations for particular sets of geodesic functions corresponding to $A_n$ and $D_n$ algebras and describe their central elements for the Poisson and quantum algebras.