The mathematics of conformal field theory: a unified approach

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This research program aims to unite three mathematical disciplines, each using its own language to rigorously describe conformal field theory, an area of physics with important applications to string theory, statistical mechanics, and condensed matter physics. The three mathematical approaches to the theory are as follows. Vertex algebras encode the symmetries of a two-dimensional field theory, while chiral algebras encode the collisions between local operators in the theory, and factorization algebras encode the quantum observables of the theory. (Observables are simply things that one can measure in the system, such as mass or velocity of particles.) Mathematically, vertex algebras are algebraic objects, while chiral and factorization algebras are objects in different flavours of geometry.   Because each of the theories approaches the physics from a different perspective, and uses a different set of mathematical techniques, they each have their own strengths and weaknesses. Although each approach is insightful on its own, it is difficult for mathematicians trained in one approach to communicate with those working with the others because they use very different mathematical languages. By building bridges between the three perspectives and combining them, this research program will allow researchers in all three fields to harness the strengths of each approach while overcoming their weaknesses, and thus will lay the foundation for a still more powerful approach to this important subject. An expected outcome of the research program is a mathematical Rosetta Stone, enabling experts in the three disciplines to communicate effectively for the first time and to join forces in their work. In the long term, this will enable me, as well as other researchers, to use new tools to solve a range of long-standing problems. Students and post-doctoral researchers trained in this program will become familiar with the wide range of mathematical techniques needed to understand all three approaches, and will be equipped to continue research in any of these three fields or in a variety of related fields.