Local index theorem for orbifold Riemann surfaces

TL;DR

This paper establishes a local index theorem for families of Cauchy-Riemann operators on orbifold Riemann surfaces, revealing how conical points influence the index and introducing a new Kähler metric on the moduli space.

Contribution

It derives a local index theorem incorporating conical points on orbifold Riemann surfaces and introduces a new elliptic Kähler metric with explicit formulas and asymptotic behavior.

Findings

Extra terms in the index theorem from conical points proportional to the symplectic form.

A simple formula for the Kähler potential of the elliptic metric.

Elliptic metric converges to the cuspidal metric as elliptic order increases.

Abstract

We derive a local index theorem in Quillen's form for families of Cauchy-Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new K\"{a}hler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local K\"{a}hler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg's zeta function.