The Gromov-Witten invariants of a compact symplectic manifold are the correlation functions of a two-dimensional topological gravity with background V. These invariants generalize such enumerative invariants of algebraic geometry as the number of curves of genus g and degree d through 3d+g-1 points in the plane. For each genus g, the genus g Gromov-Witten potential of V is the generating function of the genus g Gromov-Witten invariants. This project intends to study the differential equations satisfied by these potentials and their geometric significance. The genus zero Gromov-Witten potential satisfies the Witten-Dijkgraaf-Verlinde-Verlinde equation, which finds its geometric form in Dubrovin's theory of Frobenius manifolds. Previous work of the PI has shown that the genus one Gromov-Witten potential satisfies a differential equation defined on any Frobenius manifold. Moreover, Dubrovin and Zhang have shown that this equation has a unique solution for any semisimple Frobenius manifold. The PI proposes to investigate analogous situation for genus g greater than one and its manifold consequences.
It is well known that there is one line through two points in the plane. Similarly, there is one quadratic curve through three points in the plane. The theory of Gromov-Witten invariants is a tremendous generalization of this: one generalizes from lines to general algebraic curves, and from the plane to more general spaces. The resulting counting problems are related to the theory of such integrable systems as the Kortweg-de Vrijs equation describing waves in shallow water. In this project, following ideas of Dubrovin and Zhang, we attempt to understand this link better. Thus, this project investigates significant and surprising developments in enumerative geometry that were originally motivated physics, and potentially should cast light on a number of questions of relevance to modern physics.
