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Local index theorem for orbifold Riemann surfaces

  • Leon A. Takhtajan
  • Peter Zograf
Article

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.

Keywords

Fuchsian groups Determinant line bundles Quillen’s metric Local index theorems 

Mathematics Subject Classification

14H10 58J20 58J52 

Notes

Acknowledgements

We thank G. Freixas i Montplet for showing to us a preliminary version of [9] and for stimulating discussions. Our special thanks are to Lee-Peng Teo for carefully reading the manuscript and pointing out to us a number of misprints.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.Euler International Mathematical InstituteSaint PetersburgRussia
  3. 3.Steklov Mathematical InstituteSaint PetersburgRussia
  4. 4.Chebyshev LaboratorySaint Petersburg State UniversitySaint PetersburgRussia

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