3d Mirror Symmetry and the $\beta \gamma$ VOA

Abstract

We study the simplest example of mirror symmetry for 3d $\mathcal{N}=4$ SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello-Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the $\beta \gamma$ VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig-Kanade-McRae and Creutzig-McRae-Yang, we show that the category of line operators on the A side possesses the structure of a braided tensor category, extending the result of Allen-Wood. In addition, we prove that there is a braided tensor equivalence between the categories of line operators on the A side and B side, completing a non-trivial check of the 3d mirror symmetry conjecture. We derive explicit fusion rules as a consequence of this equivalence and obtain interesting relations with associated quantum group representations.

56 pages. v2: minor comments added, references added. Comments welcome!

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• Duality of subregular W-algebras and principal W-superalgebras
• On ribbon categories for singlet vertex algebras
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Cited by in corpus (3)

• Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs
• A non-unitary bulk-boundary correspondence: Non-unitary Haagerup RCFTs from S-fold SCFTs
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$