In this project we search for an algebraic definition that will serve the mathematician working in several of those branches touched by quantum field theory. The idea is to combine ideas from algebra and topology that serve to describe a manifold with its Poincare' duality. Applying Hochshild constructions leads to algebraic models of the free loop space of the manifold and the rich supply of operations from string topology. The BV formalism appears and solutions of the quantum master equation hopefully appear in several contexts such as differential, symplectic and holomorphic topology.
The motivation for this project at a more practical level is that several extremely interesting mathematical discussions are united in the language of theoretical physics, namely quantum field theory, but there are not precise mathematical concepts which provide such a synthesis for the working mathematician. Also once we have such underlying mathematical concepts, mathematicians will be able to develop their structure in a systematic manner and go deeper. I do not expect this work to impact the real physical examples of quantum field theory, but rather the converse-we use quantum field theory to help mathematicians by merely trying to define some part of it.
