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A Local Families Index Formula for \({\overline{\partial}}\)-Operators on Punctured Riemann Surfaces

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Abstract

Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of \({\overline{\partial}}\)-operators on the Teichmüller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space \({\mathcal{M}_{g,n}}\) in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. The degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Pierre Albin

  2. Department of Mathematics, University of Toronto, 100 st. George Street, Toronto, Ontario, MSS 363, Canada

    Frédéric Rochon

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  1. Pierre Albin
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  2. Frédéric Rochon
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Correspondence to Frédéric Rochon.

Additional information

Communicated by L. Takhtajan

The first author was partially supported by a NSF postdoctoral fellowship.

The second author was supported by a postdoctoral fellowship of the Fonds québécois de la recherche sur la nature et les technologies.

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Albin, P., Rochon, F. A Local Families Index Formula for \({\overline{\partial}}\)-Operators on Punctured Riemann Surfaces. Commun. Math. Phys. 289, 483–527 (2009). https://doi.org/10.1007/s00220-009-0816-2

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  • DOI: https://doi.org/10.1007/s00220-009-0816-2

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