Gromov-Witten theory, Donaldson-Thomas theory, and the moduli space of curves

Princeton University, Princeton NJ

Investigators

Abstract

Gromov-Witten theory is a rapidly expanding field with basic connections to many central areas of current research in mathematics and physics. The project proposed here is a wide ranging study of Gromov-Witten theory based on the techniques and discoveries of the last few years. The main topics covered are: the exact evaluations of integrals on the moduli space of curves, the proof of the universal Virasoro constraints, the establishment of the Gromov-Witten/Donaldson-Thomas correspondence, and the study of the tautological classes. These topics point in several different directions: topological string theory, integrable hierarchies, and classical algebraic geometry. Each topic is central to progress in the field, and each will be addressed with a new point of view. New methods such as localization constraints, Frobenius structures, equivariant vertex measures, and Hurwitz relations will be used. Algebraic varieties, defined by the zeros of polynomial equations, are basic objects in both classical and modern mathematics. Algebraic geometry is the study of algebraic varieties. Ideas from symplectic geometry and string theoretic physics have recently opened a new field in algebraic geometry: the study of algebraic varieties via the Gromov-Witten theory of their spaces of curves. The main objects in Gromov-Witten theory are the moduli spaces of maps from curves to algebraic varieties and their associated path integrals. The theory has been found to have remarkable structures (some proven, but most conjectural) and deep connections in many directions from quantum gravity to representation theory.