Principal Investigator: Kevin J. Costello
The 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.
Quantum field theory is the theoretical model underlying particle physics (that is, the description of the very small). Although quantum field theory has been very successful in physics, mathematicians have not been comfortable with they way quantum field theory is defined in the physics literature. The aim of this project is firstly, to further the mathematical understanding of quantum field theory; and secondly, to show how mathematical techniques can be applied to give new results about quantum field theory. The project was successful in both directions. With a collaborator, the PI has written a book (accepted for publication by Cambridge University Press) on the mathematics of quantum field theory, which will further mathematical understanding of the field. In addition, the PI has used mathematical techniques to prove new and unexpected results about particular quantum field theories called supersymmetric gauge theories. Supersymmetric gauge theories are closely related to the quantum field theories which describe the behaviour of elementary particles. Supersymmetric gauge theories can be viewed as simplified models of the more interesting (non-supersymmetric) gauge theories which model fundamental particles. The PI has developed a general link between supersymmetric gauge theories and mathematical objects called integrable systems. Optimistically, some shadow of this new understanding will apply to non-supersymmetric gauge theories as well.
