A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules

TL;DR

This paper constructs a Hermitian Topological Quantum Field Theory (TQFT) from a non-semisimple category of quantum sl(2)-modules, leading to projective mapping class group representations in indefinite unitary matrices.

Contribution

It introduces a Hermitian structure on a non-semisimple quantum sl(2) module category and proves the associated TQFT is Hermitian, extending the mathematical framework of quantum topology.

Findings

The TQFT is Hermitian.

Projective representations of the mapping class group are realized in indefinite unitary matrices.

The construction broadens the class of quantum invariants with Hermitian properties.

Abstract

We endow a non-semisimple category of modules of unrolled quantum sl(2) with a Hermitian structure. We also prove that the TQFT constructed in arXiv:1202.3553 using this category is Hermitian. This gives rise to projective representations of the mapping class group in the group of indefinite unitary matrices.