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

Abstract

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 \(\overline{U}^E(\mathfrak {gl}(1|1))\), 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 \(GL(1, {\mathbb {C}})\) connections and the resulting category of non-genuine line operators.