Orbifolds, Higher Spin Curves, and Algebraic Structures

Trustees Of Boston University, Boston

Investigators

Abstract

DMS-0204824 Takashi Kimura The goal of this proposal is to investigate geometric constructions inspired by the analogy between the moduli space of curves together with higher spin structures and the moduli space of curves together with maps into orbifolds. The former is related to higher KdV integrable hierarchies while the latter is related to the quantum cohomology of orbifolds. We will analyze the resulting algebraic structures suggested by integrable systems and the topology of these spaces. The KdV equation is a nonlinear equation modeling the behavior of water waves in a straight channel and is the first of a hierarchy of equations whose solutions are mutually compatible. Such hierarchies and their generalizations are intimately related to surfaces endowed with additional fields possessing internal symmetries. We will investigate this relationship and its ramifications.