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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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