Categorification: tangle invariants and TQFTs

TL;DR

This paper reviews categorified link invariants and TQFTs, unifying various approaches through Soergel bimodules and exploring their implications for higher-dimensional topological quantum field theories.

Contribution

It provides a unified combinatorial framework for categorified link invariants and connects them via Soergel bimodules, advancing the understanding of 2-representation theory and 4D TQFTs.

Findings

Unified categorification framework for link invariants

Connection between categorified invariants and 2-representation theory

Potential for new 4-dimensional TQFTs

Abstract

Based on different views on the Jones polynomial we review representation theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The influence of these categorifications on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categorified representations of quantum groups on the one hand and monoidal 2-categories of Soergel bimodules on the other hand might lead to new interesting 4-dimensional TQFTs.