Abstract
We evaluate half-indices of mathcal{N} = (2, 2) half-BPS boundary conditions in 3d mathcal{N} = 4 supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror transformation of a collection of half-indices for the exceptional Dirichlet boundary conditions.
Highlights
Introduction and conclusionsThere are two typical types of half-BPS boundary conditions in 3d N = 4 gauge theories
We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold
It can be constructed in the brane setup [1, 2] and has interesting dualities [3] under mirror symmetry [4,5,6]. Such a chiral BPS boundary condition admits canonical deformations that lead to the boundary Vertex Operator Algebras (VOAs) [7]. It can be further generalized by coupling to the quarter-BPS corner configurations of 4d N = 4 super Yang-Mills (SYM) theory [8,9,10,11], which played an important role in the Geometric Langlands program
Summary
It can be constructed in the brane setup [1, 2] and has interesting dualities [3] under mirror symmetry [4,5,6] Such a chiral BPS boundary condition admits canonical deformations that lead to the boundary Vertex Operator Algebras (VOAs) [7]. These limits of the N = (2, 2) half-indices can lead to the reduced indices which count the generators in the quantized Coulomb and Higgs branch algebras It has been recently argued in [26, 27] that for the boundary conditions preserving at least a maximal torus of the flavor symmetry, the two limits of the hemisphere partition function reproduce the characters of modules of the quantized Coulomb and Higgs branch algebras by checking explicitly for the exceptional Dirichlet and Verma modules in SQED and the ADHM theory. Neumann and generic Dirichlet boundary conditions which may contain line operators
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