Riemann surfaces, integrable hierarchies, and singularity theory

Abstract

In 1991, a celebrated conjecture of Witten [Wi1] asserted that the intersection theory of Deligne-Mumford moduli space is governed by KdVhierarchies. His conjecture was soon proved by Kontsevich [Ko]. Since then, the Witten-Kontsevich theorem has introduced the seemingly alien concept of integrable hierarchies to the geometry. Immediately after, a great deal of effort was spent in investigating other integrable hierarchies in GromovWitten theory. A much studied example is 2-Toda hierarchies for P1 by Okounkov-Pandharipande. It was generalized recently to orbifold P1 [J], [MT], [PR]. A famous problem of similar flavor is the Virasoro constraint for Gromov-Witten theory for an arbitrary target. The common characteristics of these problems are: (1) all of them are very difficult; (2) all of them are mysterious. In particular, the choice of integrable hierarchies seems to be matter of luck and there is no general pattern to predict the hierarchies for a given geometrical problem. Therefore, it is particularly important to explore the relation of integrable hierarchies to Gromov-Witten theory systematically. This is the main focus of this survey. In fact, this question was very much in Witten’s mind when he proposed his famous conjecture in the first place. Around the same time, he also proposed a sweeping generalization of his conjecture [Wi2, Wi3]. The core of his generalization is a remarkable first order nonlinear elliptic PDE associated to an arbitrary quasihomogeneous singularity. During the last few years, Witten’s generalization has been explored and a new Gromov-Witten type theory has been constructed by Fan-Jarvis-Ruan [FJR1, FJR2, FJR3]. In particular, Witten’s conjecture for ADE-integrable hierarchies has been verified. It is important to mention that the geometry behind these integrable