Abelian mirror symmetry of $\mathcal{N}=(2,2)$ boundary conditions

TL;DR

This paper studies mirror symmetry of boundary conditions in 3d supersymmetric Abelian gauge theories, confirming dualities of half-indices and revealing the precise transformations for exceptional boundary conditions.

Contribution

It provides a detailed analysis of Abelian mirror symmetry for boundary conditions, including the matching of half-indices and the role of elliptic stable envelopes in describing mirror transformations.

Findings

Neumann boundary condition is dual to Dirichlet in mirror theories

Half-indices match perfectly under mirror symmetry for basic boundary conditions

Elliptic stable envelope encodes the mirror transformation for exceptional boundary conditions

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.