Moduli Corrections to Gravitational Couplings from String loops

We study moduli dependent threshold corrections to gravitational couplings in the case of the heterotic string compactified on a symmetric orbifold, for untwisted moduli, extending previous analysis on gauge couplings. Like in the gauge case, the contribution comes entirely from the spacetime $N=2$ sector. As a byproduct, this calculation provides a simple derivation of the trace anomaly coefficients for the different fields coupled to gravity.