Gromov-Witten Theory and Threshold Corrections

Abstract

We present an overview of Gromov-Witten theory and its links with string theory compactifications, focussing on the GW potential as the generating function for topological string amplitudes at genus $g$. Restricting to Calabi-Yau target spaces, we give a complete derivation of the GW potential, discuss problems of multicovers and the infinite product expression. We explain the link with counting instantons or BPS states in type IIA and heterotic string theories. We show why the numbers of BPS states on the heterotic side can be a priori expressed in terms of those on the type IIA side, and vice versa. We compute heterotic one-loop integrals to obtain the genus $g$ GW potential, and detail two ways to obtain threshold corrections for heterotic orbifolds, a prerequisite for the notorious work by Harvey and Moore. We review this long and cumbersome construction in a self-contained way and make it explicit in examples of compactifications. We also develop the relation to Jacobi forms and automorphic forms, and clarify the meaning of the Gopakumar-Vafa invariants.