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

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.