Gromov-Witten invariants and symplectic reduction

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Abstract Award: DMS-0604705 Principal Investigator: Alexander Givental Gromov -- Witten invariants characterize global properties of phase spaces in Hamiltonian mechanics. It is our understanding that Gromov--Witten theory has entered a very productive period of its development. It is about to become a part of a more general framework based on ideas of conformal field theory. Many key outstanding problems of the theory such as, e.g. Virasoro conjecture, or Witten's W_n-gravity conjecture are likely to get either resolved on the basis of this framework or at least advanced to a substantial degree. Respectively, we are planning to continue our study of GW-invariants, their axiomatic structure, their generalizations, their relationships with integrable systems and symplectic field theory, methods of their computation including those associated with the mirror conjecture, and try to utilize the advantages that may come from conformal field theory. A particular problem that gave the proposal its name is a long-standing conjecture about the behavior of Gromov-Witten invariants under symplectic reduction of target spaces by circle actions. From a more general prospective, problems we deal with in our research lie on the crossroad of two major pathways in mathematics of the last two centuries. One of them is the in-depth pursuit of the intricate properties of algebraic curves --- in the form inherited from works of Gauss, Abel, Jacobi, Riemann, Klein and Poincare. The other is the broad conceptual landscaping of mathematical physics dictated by the progress of classical and quantum mechanics and often associated with the names of Hamilton, Maxwell, Gibbs, Poincare, Hilbert, Einstein and Weyl. It is string theory that in the search for the ultimate laws of nature places algebraic curves in the center of the modern landscape of fundamental physics, and generates new mathematical questions and points out plausible answers with an amazing pace and persistence. Some of the problems we work on are motivated by such questions, some others hopefully provide the answers that string theory did not really anticipate.