Moduli of Spin Curves, Geometric Vertex Algebras and Quantum Cohomology

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Abstract for DMS - 0104397. The investigator will study two groups of problems related to physical models of topological string theory and two-dimensional gravity and their interaction with geometry and other branches of mathematics. Based on physical ideas and methods, E.Witten conjectured that certain topological invariants of the moduli spaces of Riemann surfaces with higher spin structures assemble in a generating function that can be determined by an infinite system of differential equations (the so-called Gelfand-Dickey hierarchy). The goal of the first part of the proposal is to study geometric and algebraic structures related to this conjecture by using techniques from the theory of Gromov-Witten invariants and quantum cohomology. Besides exposing deep connections between seemingly unrelated areas of geometry and analysis, a goal of this part of the project is to develop tools for analyzing more general cohomological field theories of spin type and the corresponding spin analogs of quantum cohomology and Gromov-Witten invariants. The second part of the proposal is related to applications and generalizations of the so-called chiral de Rham complex, a canonical sheaf of vertex algebras introduced in joint work with Malikov and Schechtman. The main goal here is to find a rigorous geometric foundation for the physical models of two-dimensional superconformal field theory on smooth varieties and orbifolds. String theory is a proposed physical theory which attempts to unify all kinds of forces and, in particular, combines Einstein's theory of gravity with the quantum theory. Because this theory cannot yet be verified on experiments, physicists working in string theory are testing it on sophisticated mathematical models. Often the same physical quantity can be computed by more than one method which gives answers expressed in terms of very different mathematical structures. This suggests the existence of deep ties between various seemingly remote parts of mathematics and leads to formulation of non-trivial conjectures connecting different mathematical objects, usually geometric and analytical in nature. The goal of the current project is a mathematical study of some of these conjectures and interactions. Besides producing new mathematical results, it should give mathematical verification of some physical models and provide more direct links between mathematical and physical methods of analysis of these models.