Topological aspects of lattice field theory in 2 dimensions

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Fusion categories are linear monoidal categories with finitely many irreducible objects. Their theory can be thought of as a categorification of finite group theory, and their applications seem equally broad. As fully-dualizable objects in the 3-category of monoidal categories, they give rise to 3-manifold invariants via the Cobordism Hypothesis. Via surgery, they also yield knot and link invariants. On the other hand, they play a crucial role in the theory of vonNeumann algebras. For these reasons and more, a lot of research has focussed on understanding and classifying fusion categories, and significant progress has been made.Fusion categories are intriguing objects, but there is an interesting question one might ask: what if we could trade the finiteness of fusion categories for compactness? After all, the theory of compact Lie groups (especially their representation theory) is beautiful and well-studied. Is the same true one categorical level higher? Natural examples of such categories come from the categorified group rings of compact Lie groups, which play a central role in the study of Chern-Simons theory by Freed-Hopkins-Lurie-Teleman, but also representation categories of virtually abelian groups.Using the language of smooth stacks, we will study a generalisation of fusion categories with a compact manifold/orbifold of simple objects. The theory combines techniques from linear category theory and differential geometry. This promises applications to Conformal Field Theory and Topological Quantum Field Theory more generally, but also seems like a logical step to take.This project aligns with the EPSRC research area in Geometry and Topology, but has strong connections with the Mathematical Physics area, both as a source of questions and as potential target for applications.This project aligns with the EPSRC research area in Geometry and Topology, but has strong connections with the Mathematical Physics area, both as a source of questions and as potential target for applications.