@MISC{Givental_symplecticgeometry, author = {Alexander B. Givental}, title = {Symplectic geometry of Frobenius structures}, year = {} }

Share

OpenURL

Abstract

The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in two-dimensional topological field theories “coupled to gravity at the tree level”. The purpose of these notes is to reiterate and expand the viewpoint, outlined in the paper [7] of T. Coates and the author, which recasts this concept in terms of linear symplectic geometry and exposes the role of the twisted loop group L (2) GLN of hidden symmetries. We try to keep the text introductory and non-technical. In particular, we supply details of some simple results from the axiomatic theory, including a several-line proof of the genus 0 Virasoro constraints not mentioned elsewhere, but merely quote and refer to the literature for a number of less trivial applications, such as the quantum Hirzebruch–Riemann–Roch theorem in the theory of cobordism-valued Gromov–Witten invariants. The latter is our joint work in progress with Tom Coates, and we would like to thank him for numerous discussions of the subject.