Quantum Hirzebruch--Riemann--Roch Theory

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AbstractAward: DMS-1007164Principal Investigator: Alexander GiventalThe project will pursue various problems of Gromov-Witten theory, that is, the theory of topological invariants of phase spaces of Hamiltonian systems. Our research focuses on the axiomatic structure of Gromov-Witten invariants, their generalizations, their relationships with integrable systems and singularity theory, and methods of their computation, including those associated with Riemann-Roch theorems and the mirror conjecture. A central goal of this project is to resolve or advance a decade-old open problem of expressing Gromov-Witten invariants of Kahler manifolds defined as holomorphic Euler characteristics of complex vector bundles in terms of cohomological invariants of these bundles. Such expression would establish a true "quantum" analogue of the Hirzebruch-Riemann-Roch theorem. As a technical tool, the orbifold version of the classical Hirzebruch-Riemann-Roch theorem will be applied to Kontsevich's moduli spaces of stable maps. Applications of the theory to finite difference equations and integrable systems, representation theory of quantum groups, and the mirror symmetry phenomenon are expected.From a more general perspective, 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 at 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 answers that string theory did not really anticipate.