In three-dimensional gauge theories, monopole operators create and destroy
vortices. We explore this idea in the context of 3d N=4 gauge theories in the
presence of an Omega-background. In this case, monopole operators generate a
non-commutative algebra that quantizes the Coulomb-branch chiral ring. The
monopole operators act naturally on a Hilbert space, which is realized
concretely as the equivariant cohomology of a moduli space of vortices. The
action furnishes the space with the structure of a Verma module for the
Coulomb-branch algebra. This leads to a new mathematical definition of the
Coulomb-branch algebra itself, related to that of
Braverman-Finkelberg-Nakajima. By introducing additional boundary conditions,
we find a construction of vortex partition functions of 2d N=(2,2) theories as
overlaps of coherent states (Whittaker vectors) for Coulomb-branch algebras,
generalizing work of Braverman-Feigin-Finkelberg-Rybnikov on a finite version
of the AGT correspondence. In the case of 3d linear quiver gauge theories, we
use brane constructions to exhibit vortex moduli spaces as handsaw quiver
varieties, and realize monopole operators as interfaces between handsaw-quiver
quantum mechanics, generalizing work of Nakajima.