Algebraic structures in perturbative quantum field theory

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AbstractAward: DMS-1007168Principal Investigator: Kevin J. CostelloThe proposed research is concerned with a new approach to perturbative quantum field theory, based on the concept of a actorization algebra. Factorization algebras are rich algebraic objects which simultaneously generalize the notion of E-n algebra and that of chiral algebra. The aim of this proposal is to show that the techniques of perturbative renormalization allow one to construct factorization algebras starting from a Lagrangian defining a physical system. Several applications of this approach to quantum field theory are also being proposed, including one related to elliptic cohomology and the Witten genus.Quantum field theory is a fundamental tool in theoretical hysics, and underpins physicists current understanding of the behaviour of elementary particles. Quantum field theory has additionally had a great influence on the development of mathematics. The aim of this research proposal is to provide mathematical foundations for quantum field theory, and to apply these foundations to problems in geometry related to physics.