Topological aspects of lattice field theory in 2 dimensions

融合范畴是具有有限多个不可约对象的线性么半群范畴。它们的理论可以被认为是有限群论的一个范畴，它们的应用似乎也同样广泛。作为么半群范畴3-范畴中的完全对偶化对象，它们通过协边假设产生了3-流形不变量。通过手术，它们还产生了纽结和链接不变量。另一方面，它们在vonNeumann代数理论中起着至关重要的作用。由于这些原因和更多的原因，许多研究都集中在对融合范畴的理解和分类上，并取得了重大进展。融合范畴是有趣的对象，但人们可能会问一个有趣的问题：如果我们可以用融合范畴的有限来换取紧致性，会怎么样？毕竟，紧李群的理论(特别是它们的表示理论)是美丽的，也是研究得很好的。同样的真理是不是高出一个分类级别？这类范畴的自然例子来自紧致李群的范畴化群环，它在Freed-Hopkins-Lurie-Teleman的Chern-Simons理论的研究中起着核心作用，但也是虚拟阿贝尔群的表示范畴。利用光滑栈的语言，我们将研究具有紧致单对象流形/或分支的融合范畴的推广。该理论结合了线性范畴理论和微分几何的技术。这个项目与EPSRC几何与拓扑学的研究领域相一致，但作为一个潜在的应用目标，它与数学物理领域有很强的联系。这个项目与EPSRC在几何与拓扑学的研究领域相一致，但作为潜在的应用目标，它与数学物理领域有很强的联系。

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.