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It is known that the category $\mathscr{C}^H$ of nilpotent weight modulesover the quantum group associated with the super Lie algebra$\mathfrak{sl}(2|1)$ is a relative pre-modular $G$-category. Its modified traceenables to define an invariant of $3$-manifolds. In this article we show thatthe category $\mathscr{C}^H$ is a relative modular $G$-category which allowsone to construct a family of non-semi-simple extended topological quantum fieldtheories which surprisingly are anomaly free. The quantum group associated with$\mathfrak{sl}(2|1)$ is considered at odd roots of unity.

Anomaly-free TQFTs from the super Lie algebra $\mathfrak{sl}(2|1)$