Narrow Results

It is known that the category ${\mathcal{𝒞}}^H$ of nilpotent weight modules over the quantum group associated with the super Lie algebra $𝔰𝔩(2|1)$ is a relative pre-modular $G$-category. Its modified trace enables one to define an invariant of $3$-manifolds. In this paper, we show that the category ${\mathcal{𝒞}}^H$ is a relative modular $G$-category which allows one to constructing a family of non-semi-simple extended topological quantum field theories, and the family of TQFTs is anomaly free. The quantum group associated with $𝔰𝔩(2|1)$ is considered at odd roots of unity.