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 $\mathfrak{gl}(1|1)$. Our proposal is that the category of line operators $\mathcal{C}$ can be identified with the derived category of modules for a boundary vertex operator algebra $\mathcal{V}$ realized as a certain infinite-order simple current extension of the affine current algebra $V(\mathfrak{gl}(1|1))$ by boundary monopole operators. By translating this simple current extension of $V(\mathfrak{gl}(1|1))$ to the unrolled, restricted quantum group $\overline{U}^E(\fgl(1|1))$, we show that our category of line operators admits a second description in terms of a quasi-quantum group $\mathcal{A}$ 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.