Line Operators in U(1|1) Chern–Simons Theory

We analyze the non-semisimple category of line operators in Chern–Simons gauge theories based off the Lie superalgebra \ (gl (1|1) \). Our proposal is that the category of line operators \ (C\) can be identified with the derived category of modules for a boundary vertex operator algebra \ (V\) realized as a certain infinite-order simple current extension of the affine current algebra \ (V (gl (1|1) ) \) by boundary monopole operators. By translating this simple current extension of \ (V (gl (1|1) ) \) to the unrolled, restricted quantum group \ (UE (gl (1|1) ) \), we show that our category of line operators admits a second description in terms of a quasi-quantum group \ (A\) realized by uprolling. We also compare our results across an expected physical duality with the cyclic orbifold of a free, B -twisted hypermultiplet and find a slight discrepancy at the level of braiding and associator. We end with a detailed analysis of coupling to background flat \ (GL (1, {C}) \) connections and the resulting category of non-genuine line operators.