A local families index formula for d-bar operators on punctured Riemann surfaces

TL;DR

This paper derives a local index formula for families of d-bar operators on punctured Riemann surfaces, connecting heat kernel methods with geometric invariants and curvature computations.

Contribution

It introduces a new local index formula applicable to families of d-bar operators on punctured Riemann surfaces, linking heat kernel techniques with moduli space geometry.

Findings

Derived a local index formula using heat kernel methods

Expressed the formula in terms of Mumford-Morita-Miller classes

Calculated the curvature of the determinant line bundle with Quillen connection

Abstract

Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of d-bar operators on the Teichmuller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space 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.