Abstract: Using heat kernel methods developed by Vaillant, we will show how to obtain a local index formula for families of d-bar operators parametrized by the Teichmuller space of Riemann surfaces of genus g with n punctures. The formula also holds on the corresponding moduli space in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. As we will indicate, 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.