| Résume |
In physics, 3d mirror symmetry is a duality of topological twists of 3d quantum field theories with N=4 supersymmetry. A consequence of this duality is an isomorphism between symplectic varieties of the mirror pairs of theories called the Higgs and Coulomb branches. In work of Costello and Gaiotto, a vertex algebra is constructed on holomorphic boundary conditions of the twists of these theories and it was conjectured that the associated variety and derived endomorphism algebras of this boundary vertex algebra can recover the Higgs and Coulomb branches. In this talk I will discuss joint work with Andrea Ferrari in which we prove that the boundary vertex (super)algebra for the abelian gauge theory with gauge group U(1) acting on n>2 hypermultiplets is isomorphic to the simple quotient of the psl(n|n) vertex superalgebra at level 1, and that the associated variety of this vertex superalgebra is isomorphic to the closure of the minimal nilpotent orbit of sl(n). In doing so, we prove the 3d Higgs branch conjecture for this set of examples.
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