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 \(Ū^E(gl(1|1))\) , 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.