Abelian dualities of $\mathcal{N}=(0,4)$ boundary conditions

We propose dual pairs of $\mathcal{N}=(0,4)$ half-BPS boundary conditions for 3d $\mathcal{N}=4$ Abelian gauge theories related to mirror symmetry and S-duality by showing the matching of boundary 't Hooft anomalies and supersymmetric indices. We find simple $\mathcal{N}=(0,4)$ mirror symmetry between 2d $\mathcal{N}=(0,4)$ Abelian gauge theories and free Fermi multiplets that generalizes $\mathcal{N}=(0,2)$ Abelian duality. We also propose a prescription for computing half-index of enriched Neumann boundary condition including 2d boundary bosonic matters by gauging the 2d boundary flavor symmetry of Dirichlet boundary condition. By coupling $\mathcal{N}=(0,4)$ half-BPS boundary configurations of 3d $\mathcal{N}=4$ gauge theories to quarter-BPS corner configurations of 4d $\mathcal{N}=4$ Super Yang-Mills theories, we further obtain a new type of 4d-3d-2d duality that may involve 3d non-Abelian gauge symmetry.