Principal Investigator: Ezra Getzler
In this proposal, we study an important mathematical problem suggested by string theory: the construction of a theory of secondary Gromov-Witten invariants. In other words, an analogue of rational homotopy theory for Floer homology. While the ideas of rational homotopy theory have been extensively applied in algebraic topology, they have not received much attention in the study of Gromov-Witten invariants, or more generally, of topological field theory; this proposal attempts to rectify this. We will examine whether an analogue of formality holds for compact Kaehler targets. Our results will be formulated in terms of our theory of homotopy Gerstenhaber algebras, and Tamarkin and Tsygan's theory of homotopy Batalin-Vilkovisky algebras.
In a sense, this proposal addresses the difficult problem of understanding the mathematical formulation of string theory, which is itself a very promising descriptions of the fundamental laws of nature. Throughout its forty year history, string theory has drawn on the insights of modern mathematics. But there is a sense among researchers that the final formulation of the theory has not yet been attained: we believe that this proposal will bring us closer to finding the mathematical language which is needed for such a description.
