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We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto--Witten theory with linear matter. Our constructions are based on relative modular structures on the category of weight modules over an unrolled quantization of a Lie superalgebra. The Lie superalgebra, originally defined by Gaiotto and Witten, is associated to a complex symplectic representation of a metric abelian Lie algebra. The physical theories we model admit alternative realizations as Chern--Simons-Rozansky--Witten theories and supergroup Chern--Simons theories and include as particular examples global forms of gl(1,1)-Chern--Simons theory and toral Chern--Simons theory. Fundamental to our approach is the systematic incorporation of non-genuine line operators which source flat connections for the topological flavour symmetry of the theory. 

We introduce an unrolled quantization U_q^E(gl(1,1)) of the complex Lie superalgebra gl(1,1) and use its categories of weight modules to construct and study new three dimensional non-semisimple topological quantum field theories. These theories are defined on categories of cobordisms which are decorated by ribbon graphs and cohomology classes and take values in categories of graded super vector spaces. Computations in these theories are enabled by a detailed study of the representation theory of U_q^E(gl(1,1)). We argue that by restricting to subcategories of integral weight modules we obtain topological quantum field theories which are mathematical models of Chern--Simons theories with gauge supergroups psl(1,1,) and U(1,1) coupled to background flat \mathbb{C}^{\times}-connections, as studied in the physics literature by Rozansky--Saleur and Mikhaylov. In particular, we match Verlinde formulae and mapping class group actions on state spaces of non-generic tori with results in the physics literature. We also obtain explicit descriptions of state spaces of generic surfaces, including their graded dimensions, which go beyond results in the physics literature. 

We construct a two dimensional unoriented open/closed topological field theory from a finite graded group $$\pi:\Gh \twoheadrightarrow \{1,-1\}$$, a $$\pi$$-twisted $$2$$-cocycle $$\hat{\theta}$$ on $$B \hat{G}$$ and a character $$\lambda: \hat{G} \rightarrow U(1)$$. The underlying oriented theory is a twisted Dijkgraaf--Witten theory. The construction is based on a detailed study of the $$(\hat{G}, \hat{\theta},\lambda)$$-twisted Real representation theory of $$\textnormal{ker} \pi$$. In particular, twisted Real representations are boundary conditions of the unoriented theory and the generalized Frobenius--Schur element is its crosscap state.