Topological aspects of lattice field theory in 2 dimensions

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute

Abstract

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.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509711/1 01/10/2016 30/09/2021
1941778 Studentship EP/N509711/1 01/10/2017 31/03/2021 Christoph Weis
 
Description Orbifold Tensor Categories were defined and studied. These categories are generalisations of fusion categories, and promise to have application in Conformal Field Theory and Topological Quantum Field Theories more generally. Initial results include a local-to-global principle for rigidity, leveraging the interplay between differential geometry and linear category theory in this framework. There has also been progress towards a local classification of Orbifold Tensor Categories.
Exploitation Route Given that examples of Orbifold Tensor categories come up in Conformal Field Theory, toral Chern-Simons-Theory and Representation Theory (amongst others), a systematic study could provide further insight into these areas of research.
Sectors Other