B-Twisted Gaiotto–Witten Theory and Topological Quantum Field Theory

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.