Kazhdan‐Lusztig correspondence for vertex operator superalgebras from abelian gauge theories

Abstract We prove the Kazhdan–Lusztig correspondence for a class of vertex operator superalgebras that, via the work of Costello–Gaiotto, arise as boundary vertex operator algebra (VOAs) of the topological B twist of 3d abelian gauge theories. This means that we show equivalences of braided tensor categories of modules of certain affine vertex superalgebras and corresponding quantum supergroups. We build on the work of Creutzig–Lentner–Rupert for this large class of VOAs and extend it since, in our case, the categories do not have projective objects, and objects can have arbitrary Jordan–Hölder length. Our correspondence significantly improves the understanding of the braided tensor category of line defects associated with this class of topological quantum field theory (TQFT) by realizing line defects as modules of a Hopf algebra. In the process, we prove a special case of the conjecture of Semikhatov–Tipunin, relating logarithmic conformal field theory (CFTs) to Nichols algebras of screening operators.