On the quantum cohomology of a symmetric product of an algebraic curve

Abstract

The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example, as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interval [(3/4)g, g−1). Otherwise, they still give a formula modulo third-order terms. Explicit generators and relations are also given unless d is in [(4/5)g−3/5, g−1). The virtual class on the space of stable maps plays a significant role. But the central ideas ultimately come from Brill-Noether theory: specifically, a formula of J. Harris and L. Tu for the Chern numbers of determinantal varieties. The case of d=g−1 is especially interesting: it resembles that of a Calabi-Yau 3-fold, and the Aspinwall-Morrison formula enters the calculations. A detailed analogy with A. Givental's work is also explained.