  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 $\mathfrak{gl}(1 \vert
1)$-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.
