  We study moduli spaces of twisted quasimaps to a hypertoric variety $X$,
arising as the Higgs branch of an abelian supersymmetric gauge theory in three
dimensions. These parametrise general quiver representations whose building
blocks are maps between rank one sheaves on $\mathbb{P}^1$, subject to a
stability condition, associated to the quiver, involving both the sheaves and
the maps. We show that the singular cohomology of these moduli spaces is
naturally identified with the Ext group of a pair of holonomic modules over the
`quantized loop space' of $X$, which we view as a Higgs branch for a related
theory with infinitely many matter fields. We construct the coulomb branch of
this theory, and find that it is a periodic analogue of the coulomb branch
associated to $X$. Using the formalism of symplectic duality, we derive an
expression for the generating function of twisted quasimap invariants in terms
of the character of a certain tilting module on the periodic coulomb branch. We
give a closed formula for this generating function when $X$ arises as the
abelianisation of the $N$-step flag quiver.
