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We introduce an unrolled quantization $U_q^E(\mathfrak{gl}(1 \vert 1))$ ofthe complex Lie superalgebra $\mathfrak{gl}(1 \vert 1)$ and use its categoriesof weight modules to construct and study new three dimensional non-semisimpletopological quantum field theories. These theories are defined on categories ofcobordisms which are decorated by ribbon graphs and cohomology classes and takevalues in categories of graded super vector spaces. Computations in thesetheories are enabled by a detailed study of the representation theory of$U_q^E(\mathfrak{gl}(1 \vert 1))$, both for generic and root of unity $q$. Weargue that by restricting to subcategories of integral weight modules we obtaintopological quantum field theories which are mathematical models ofChern--Simons theories with gauge supergroups $\mathfrak{psl}(1 \vert 1)$ and$\mathfrak{gl}(1 \vert 1)$ coupled to background flat$\mathbb{C}^{\times}$-connections, as studied in the physics literature byRozansky--Saleur and Mikhaylov. In particular, we match Verlinde formulae andmapping class group actions on state spaces of non-generic tori with results inthe physics literature. We also obtain explicit descriptions of state spaces ofgeneric surfaces, including their graded dimensions, which go beyond results inthe physics literature.

Three dimensional topological quantum field theory from $U_q(\mathfrak{gl}(1 \vert 1))$ and $U(1 \vert 1)$ Chern--Simons theory