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.

arXiv comment56 pages. v2: minor comments added, references added. Comments welcome!

References in corpus (16)

• Vertex Algebras and Algebraic Curves
• Mirror Symmetry in Three Dimensional Gauge Theories
• Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes
• Mirror Symmetry in Three-Dimensional Gauge Theories, SL(2,Z) and D-Brane Moduli Spaces
• Dual Boundary Conditions in 3d SCFT's
• Schur-Weyl Duality for Heisenberg Cosets
• Vertex Operator Algebras and 3d N=4 gauge theories
• Simple current extensions beyond semi-simplicity
• Bosonic Ghosts at $c=2$ as a Logarithmic CFT
• Mirror symmetry and line operators
• Duality of subregular W-algebras and principal W-superalgebras
• On fusion rules and intertwining operators for the Weyl vertex algebra
• On ribbon categories for singlet vertex algebras
• Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion
• Direct limit completions of vertex tensor categories
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$

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 $A$