The non semi-simple TQFT of the sphere with four punctures

TL;DR

This paper computes and proves the faithfulness of the non semi-simple TQFT representation of the mapping class group of a four-punctured sphere, and compares it with braid group representations.

Contribution

It provides the first explicit computation of the non semi-simple TQFT representation for this case and demonstrates its faithfulness.

Findings

Representation is faithful for the four-punctured sphere

Comparison with braid group representations reveals similarities and differences

Advances understanding of non semi-simple TQFTs in low-genus surfaces

Abstract

In this work, we compute the representation of the mapping class group of the sphere with $4$ punctures arising from the non semi-simple TQFT (constructed by Blanchet--Costantino--Geer--Patureau). We show that it is faithful. Lastly, we compare quantum representations of punctured spheres in general with those of braid groups.