Mathematical Problems Inspired by String Theory

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The investigator studies IIA and IIB superconformal field theories on a Calabi-Yau variety in an attempt to give a rigorous mathematical definition of the vertex algebra of the theories. This algebra is expected to be a deformation of the cohomology of the chiral de Rham complex constructed by Malikov, Schechtman and Vaintrob. The second aim of the project is to extend the definition of IIA and IIB models to some singular varieties. The most pressing question is to connect two existing definitions of elliptic genus of singular varieties, which is a generalization of cohomological McKay correspondence. The third part of the project focuses on problems from number theory that arise in connection with the concept of elliptic genus and are related to products of Eisenstein series. String theory in its various flavors is a leading candidate for the ``theory of everything'' in mathematical physics. Its development has led to rapid growth in several areas of mathematics, in particular algebraic geometry, which concerns itself with spaces of solutions of polynomial equations. Unfortunately, there is still a lack of rigorous understanding of some of the underlying rich mathematical structures, which the project aims to rectify. Such mathematical understanding is important, because it may be a necessity for future development of string theory which in turn may yield a more coherent picture of the basic processes and forces of our universe. A separate part of the project deals with certain number-theoretical questions inspired by string theory. These are related to long-standing mathematical problems in the theory of elliptic curves. The theory of elliptic curves is a crucial part of the proof of Fermat's Last Theorem and has practical applications to cryptography. This project is jointly funded by the Algebra, Number Theory, and Combinatoric Program and the Topology and Geometric Analysis Program.