The geometry of topological quantum field theories (ERC-StG)

The project is devoted to a bottom-up approach to the geometry of topological quantum field theories (TQFTs).

TQFTs are useful for various reasons: Firstly and originally, they can be viewed as toy models of string theory, dramatically simplifying the latter up to solvability. Secondly and even better, full-fledged superstring theories, which are usually built on a Calabi-Yau "target space" X, can be projected onto TQFTs, yielding a selection of original superstring data prone to simplified calculations. These data, in fact, are often not even accessible otherwise. Since they are closely related to the underlying target space X, such calculations regularly have an important impact on geometry.

The research project is based on fundamental concepts concerning the very geometry of moduli spaces of TQFTs, and it aims at a broad view of TQFTs, including D-branes and the role of generalized theta functions as well as BPS algebras and automorphic forms. Following a mathematical route we will first develop a complete understanding of the geometric properties of moduli spaces of TQFTs. As a starting point, Hertling's "TERP structures" yield an abstract description of such moduli spaces, while TQFTs shall be viewed as arising from quantization of spaces with TERP structure. The approach combines the advantages of both a mathematician's and a physicist's viewpoint: It puts the proposed research on a solid mathematical foundation while, by exploiting their common roots in physics, it relates seemingly disjoint areas of mathematics which have evolved over a period of more than twenty years.