Algebraic Structures of Mathematical Physics

Abstract

Award: DMS-0805785

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.