Local index theorem for orbifold Riemann surfaces

L.A. Takhtajan, Peter Zograf

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    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.

    Original languageEnglish
    Pages (from-to)1119-1143
    Number of pages25
    JournalLetters in Mathematical Physics
    Volume109
    Issue number5
    DOIs
    Publication statusPublished - 1 May 2019

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    Scopus subject areas

    • Statistical and Nonlinear Physics
    • Mathematical Physics

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