Gromov - Witten invariants and integrable hierarchies

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Abstract Award: DMS-0306316 Principal Investigator: Alexander Givental Gromov-Witten invariants characterize global properties of phase spaces in Hamiltonian mechanics. The project concerns with various problems of constructing and generalizing Gromov-Witten invariants, developing methods of their computation, interpreting the geometrical information they encode and understanding the rules that govern relationships among them. In particular, the role of integrable hierarchies and their Virasoro symmetries in Gromov-Witten theory and in its mirror partner - singularity theory - is to be investigated. To this end, certain vertex operator constuctions based on the theory of vanishing cycles in singularity theory will be explored as a promising tool for describing and manipulating with integrable hierarchies. Exploiting the twisted loop group of hiden symmetries discovered recently in the structure of Gromov-Witten invariants and uncovering the origin of the symmetries are among key motives for this project. In a broader prospective, the projects continues the legacy of the past two centures in the development and application of modern mathematics. The methods of algebraic geometry, topology, group theory and quantum mechanics laid down in the classical work of Gauss, Riemann, Lie, Klein, Poincare and Weyl come together to address the problems posed lately by string theory in its search for the ultimate unity of fundamental interactions.