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Local index theorem for orbifold Riemann surfaces

Leon A. Takhtajan, Peter Zograf

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 Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler 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.

Local index theorem for orbifold Riemann surfaces

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 Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler 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.

Paper Structure

This paper contains 12 sections, 5 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic plane and Fuchsian groups
  4. Laplacians on Riemann orbisurfaces
  5. Deformation theory
  6. Main results
  7. Connection form on the determinant bundle
  8. The first Chern form
  9. Concluding remarks
  10. Local potential for elliptic metric
  11. Cuspidal and elliptic metrics
  12. Elliptic metric and Selberg zeta values

Key Result

Lemma 1

Let be the Fourier series of the function $f_{\mu\bar{\nu}}(u)$ on $\mathbb{D}$. Then

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Remark 1
  • Theorem \oldthetheorem
  • Remark 2
  • proof
  • Lemma 2
  • proof
  • Theorem \oldthetheorem
  • Remark 3
  • ...and 2 more