Monoidal Categories of Line Operators and 3d N = 4 Mirror Symmetry

We study the simplest example of 3d mirror symmetry for N = 4 supersymmetric gauge theories: the A-twist of a free hypermultiplet and the B-twist of U(1) gauge theory coupled to a single hypermultiplet (SQED[1]). Our interest is primarily directed towards the category of line operators in each theory as well as the tensor structure they each possess. After reviewing these topics in a general setting, we return to our main example and identify the categories of line operators therein as appropriate module subcategories of certain vertex operator algebras; our approach is analogous to that used by Witten and uses results of Costello-Gaiotto. One vertex operator algebra that appears is the familiar beta-gamma system, the second one is related to the affine superalgebra \widehat{gl(1|1)}. We provide an explicit description of these module categories, which are strictly larger than those previously studied by Ridout-Wood and Allen-Wood. We additionally prove that these module categories are equivalent as braided tensor categories, culminating in an equivalence between the categories of line operators of original interest. This result completes a nontrivial check of the 3d mirror symmetry conjecture. We compute the tensor structure induced on the category of beta-gamma modules from this equivalence, extending the work of Allen-Wood. We finally comment on work in progress generalizing this equivalence to theories with arbitrary abelian gauge groups