Local index theorem for orbifold Riemann surfaces
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.
KeywordsFuchsian groups Determinant line bundles Quillen’s metric Local index theorems
Mathematics Subject Classification14H10 58J20 58J52
We thank G. Freixas i Montplet for showing to us a preliminary version of  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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