Mathematical Problems Inspired by String Theory

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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. Thethird part of the project focuses on problems from number theory thatarise in connection with the concept of elliptic genus and are relatedto 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 particularalgebraic 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 isimportant, because it may be a necessity for future development of string theory which in turn may yield a more coherent picture of thebasic 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 mathematicalproblems in the theory of elliptic curves. The theory of ellipticcurves 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.