This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
In this proposal, the proposer intends to study universal equations and other properties of Gromov-Witten invariants of compact symplectic manifolds, as well as their interaction with integrable systems. Gromov-Witten invariants are defined by the intersection theory on moduli spaces of stable pseudo-holomorphic maps from Riemann surfaces to compact symplectic manifolds. They have intuitive interpretation of counting pseudo-holomorphic curves in compact symplectic manifolds. Universal equations are partial differential equations which govern fundamental behavior of Gromov-Witten invariants of all compact symplectic manifolds, as well as for intersection numbers on moduli spaces of higher spin curves. They provide powerful machineries in the computation of these invariants and also play important roles in the study of the Virasoro conjecture of Eguchi-Hori-Xiong and S. Katz. The Virasoro conjecture is a generalization of Witten's KdV conjecture (proved by Kontsevich) and serves as a bridge between Gromov-Witten theory and integrable systems. The study of universal equations also helps the understanding of structures of tautological rings of moduli spaces of curves.
The theory of Gromov-Witten invariants has important applications in symplectic geometry, algebraic geometry, gauge theory, and integrable systems. The proposed work concerns the fundamental structures of this theory. Besides its applications in mathematics, the Gromov-Witten theory corresponds to topological sigma models in string theory. Therefore the proposed project also has strong connections with theoretic physics. Some of the deepest conjectures in this field, including some conjectures in this proposal, actually came from physicists. The resolution of problems proposed here will give a rigorous support to intuitions of physicists and helps the understanding of the topological sigma model in string theory.
