Two primary schools putting QFT on firm mathematical footing are Algebraic Quantum Field Theory (AQFT) and Functorial Quantum Field Theory (FQFT). These are analogous to the Heisenberg and Schrödinger pictures of quantum mechanics: the former focuses on observables and their algebraic properties, while the latter describes the (time) evolution of state spaces. The two approaches were studied independently until very recently, when attempts began to relate them.
My doctoral work concerns a special subclass of quantum field theories: two-dimensional chiral conformal field theories (CFTs). Two-dimensional CFTs describe critical phenomena in various condensed matter systems, and they also form the building blocks of string theory. The aim is to give a construction of a fully extended Segal chiral CFT (a functor defined on the category of conformal manifolds) from the algebraic data of conformal nets.
I am also working to give a rigorous mathematical basis to the idea that a d-dimensional (T)QFT has a monoidal (d–1)-category of symmetries. Physicists are discovering global and higher categorical symmetries in QFTs, and my project aims to show that such structures are expected to arise when symmetries are generalised to topological interfaces between QFTs. This involves constructing an (∞,n)-category Bord, with topological defects of all codimensions.
Over the past decade, higher categories have proven to be an essential ingredient in studying quantum field theories from a mathematical perspective. Casting QFT in the language of higher category theory naturally and elegantly incorporates many subtle concepts, including generalised symmetries, topological defects, and locality.
My research interests include the Stolz–Teichner conjecture, the chiral free fermion, higher von Neumann algebras, higher Hilbert spaces, and more generally the higher categorical constructions that capture the structure of the theory of operator algebras.