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.

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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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Correspondence to Peter Zograf.

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Supported by RScF Grant 16-11-10039.

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Takhtajan, L.A., Zograf, P. Local index theorem for orbifold Riemann surfaces. Lett Math Phys 109, 1119–1143 (2019). https://doi.org/10.1007/s11005-018-01144-w

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Keywords

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

Mathematics Subject Classification

  • 14H10
  • 58J20
  • 58J52