Anomaly-free TQFTs from the super Lie algebra $\mathfrak{s}\mathfrak{l}(2\mathrm{\mid }1)$

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