Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups

TL;DR

This paper develops a categorical framework for Hermitian non-semisimple TQFTs, linking Hermitian modular categories to modified TQFTs and introducing non-semisimple Levin-Wen models with potential applications in topological phases.

Contribution

It introduces a new Hermitian categorical framework for non-semisimple TQFTs and constructs novel Levin-Wen models based on quantum supergroups.

Findings

Hermitian modular categories lead to modified Hermitian TQFTs

Examples derived from quantum groups and superalgebras

New pseudo-Hermitian topological phases proposed

Abstract

We develop the categorical context for defining Hermitian non-semisimple TQFTs. We prove that relative Hermitian modular categories give rise to modified Hermitian WRT-TQFTs and provide numerous examples of these structures coming from the representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev-Viro TQFT is applied to define new pseudo-Hermitian topological phases that can be considered as non-semisimple analogs of Levin-Wen models.