Geometry and topological field theories (Subactivity of WAG 07-08)

This is a three month subactivity of the Warwick Algebraic Geometry Symposiums 2007/08, which will be devoted to questions coming from topological quantum field theory. In particular, this subactivity of WAG07-08 will host an international spring school and a conference.

Ideas from quantum field theory and string theory have had an enormous impact on geometry over the last 20 years; the most obvious links for algebraic geometers relate to mirror symmetry, Gromov-Witten invariants and the McKay correspondence. The topics we want to focus on during WAG07-08 go back to papers of Cecotti and Vafa and their coauthors around 1991. These papers introduced a whole series of ideas originating in QFT, that have proved to be basic new ingredients in mathematics. The spectrum of ideas we will explore range from the physics of string theory, topological QFT, the formalism of tt* equations and their classical limits, to mathematical rigorous notions such as Dubrovin's Frobenius structures on manifolds and Hertling's TERP structures. A modest down-to-earth view of this is as an enrichment of Hodge structures (and their variations) to include data from the symplectic side of mirror symmetry; as well as many predictions in the realm of enumerative geometry, this gives concrete results in applications to singularity theory and algebraic geometry, and relates directly to problems in geometry such as moduli of flat structures on bundles and integrable systems.