Topological Field Theory, Higher Categories, and Their Applications

TL;DR

This paper discusses the role of higher categories in formulating extended topological field theories (TFTs) in higher dimensions, highlighting recent physical interpretations and applications in geometry and representation theory.

Contribution

It outlines new applications of higher-dimensional TFTs, including classification of monoidal deformations and insights into geometric Langlands duality.

Findings

Concrete examples of extended TFTs like Rozansky-Witten model are constructed.

Higher categorical structures have meaningful physical interpretations.

Applications include classification of monoidal deformations and geometric Langlands duality.

Abstract

It has been common wisdom among mathematicians that Extended Topological Field Theory in dimensions higher than two is naturally formulated in terms of n-categories with n> 1. Recently the physical meaning of these higher categorical structures has been recognized and concrete examples of Extended TFTs have been constructed. Some of these examples, like the Rozansky-Witten model, are of geometric nature, while others are related to representation theory. I outline two application of higher-dimensional TFTs. One is related to the problem of classifying monoidal deformations of the derived category of coherent sheaves, and the other one is geometric Langlands duality.