Topological aspects of lattice field theory in 2 dimensions

融合范畴是具有有限多不可约对象的线性一元范畴。他们的理论可以被认为是有限群论的一个范畴，他们的应用似乎同样广泛。作为一元范畴的三范畴中的完全对偶对象，它们通过协点假设得到三流形不变量。通过手术，它们也会产生结和连接不变量。另一方面，它们在冯诺伊曼代数理论中起着至关重要的作用。由于这些原因和其他原因，许多研究集中在理解和分类聚变类别上，并取得了重大进展。融合范畴是一个有趣的对象，但有一个有趣的问题可能会问：如果我们可以用融合范畴的有限性来换取紧凑性呢？毕竟，紧李群的理论（尤其是它们的表示理论）是美丽的，而且研究得很好。同样的真理是否更高一个分类层次？这类范畴的自然例子来自紧李群的有分类群环，它在Freed-Hopkins-Lurie-Teleman的chen - simons理论研究中起着核心作用，但也是虚拟阿贝尔群的表示范畴。使用光滑堆栈语言，我们将研究具有简单对象的紧流形/轨道的融合范畴的推广。该理论结合了线性范畴理论和微分几何的技术。这为保形场论和拓扑量子场论提供了更广泛的应用前景，但也似乎是一个合乎逻辑的步骤。该项目与EPSRC的几何和拓扑研究领域保持一致，但与数学物理领域有着密切的联系，既是问题的来源，也是应用的潜在目标。该项目与EPSRC的几何和拓扑研究领域保持一致，但与数学物理领域有着密切的联系，既是问题的来源，也是应用的潜在目标。

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.