Principal Investigator: Alexander A. Voronov
The goal of the project is to solve a number of important problems in the rapidly developing fields of Topological Field Theory (TFT), Symplectic Field Theory (SFT), and Gromov-Witten theory. The first part of the project will span from higher category theory, to cobordisms and to quantum field theories. The plan is to place cobordisms of manifolds with corners within an appropriate n-category framework and describe TFTs as n-functors from the n-category of cobordisms to that of n-vector spaces, as well as show that physical models, such as gauge (Wess-Zumino-Witten), Yang-Mills, Chern-Simons, Seiberg-Witten theories, and sigma-model may be described as such higher TFTs. The second part of the project consists in bringing together algebraic geometric and symplectic methods to construct a full solution to the so-called Quantum Master Equation in Gromov-Witten theory. This equation describes the topology of the moduli spaces of holomorphic curves and relevant algebraic structures, providing important invariants in symplectic and algebraic geometry. The third part of the project aims at lifting Gromov-Witten theory to the (Floer) chain level and developing a combinatorial version of Gromov-Witten theory, thus bridging the areas of enumerative algebraic geometry, symplectic Floer theory, and graph homology. The last, SFT part of the project will result in constructing a new compactification of the moduli space of Riemann surfaces, which would govern the algebraic operations and invariants arising in SFT. This compactification will be an SFT analogue of the Deligne-Mumford compactification relevant to Gromov-Witten theory.
The project aims at discovering and studying new algebraic structures in topology suggested or motivated by mathematical physics, in particular, string theory, Symplectic Field Theory, and Gromov-Witten theory. Another long-term goal is to build a bridge between several mathematical cultures working on problems related to mathematical physics. These cultures include algebraists, algebraic topologists, symplectic geometers, algebraic geometers, and geometric topologists, to name a few. The algebraic structures is a mathematical reincarnation of such fundamental structures of physical theories as correlators and relations between them (Ward identities). Understanding this structure is crucial for understanding the physical theory. From the point of view of mathematics, the project leads to new mathematical ideas, new algebra, geometry, and topology, motivated by physics.
