Blocks and Vortices in the 3d ADHM Quiver Gauge Theory

We study the hemisphere partition function of a three-dimensional $\mathcal{N}=4$ supersymmetric $U(N)$ gauge theory with one adjoint and one fundamental hypermultiplet -- the ADHM quiver theory. In particular, we propose a distinguished set of UV boundary conditions which yield Verma modules of the quantised chiral rings of the Higgs and Coulomb branches. In line with a recent proposal by two of the authors in collaboration with M. Bullimore, we show explicitly that the hemisphere partition functions recover the characters of these modules in two limits, and realise blocks gluing exactly to the partition functions of the theory on closed three-manifolds. We study the geometry of the vortex moduli space and investigate the interpretation of the vortex partition functions as equivariant indices of quasimaps to the Hilbert scheme of points in $\mathbb{C}^2$. We also investigate half indices of the ADHM quiver gauge theory in the presence of a line operator and discuss their geometric interpretation. Along the way we find interesting relations between our hemisphere blocks and related quantities in topological string theory and equivariant quantum K-theory.