Vertex Operator Algebras and 3d N $$ \mathcal{N} $$ = 4 gauge theories

Abstract We introduce two mirror constructions of Vertex Operator Algebras associated to special boundary conditions in 3d N $$ \mathcal{N} $$ = 4 gauge theories. We conjecture various relations between these boundary VOA’s and properties of the (topologically twisted) bulk theories. We discuss applications to the Symplectic Duality and Geometric Langlands programs