  This paper is concerned with a non-compact GIT quotient of a vector space, in
the presence of an abelian group action and an equivariant regular function
(potential) on the quotient. We define virtual counts of quasimaps from
prestable curves to the critical locus of the potential, and prove a gluing
formula in the formalism of cohomological field theories.
  The main examples studied in this paper is when the above setting arises from
quivers with potentials, where the above construction gives quantum correction
to the equivariant Chow homology of the critical locus. Following similar ideas
as in quasimaps to Nakajima quiver varieties studied by the Okounkov school, we
analyse vertex functions in several examples, including Hilbert schemes of
points on $\mathbb{C}^3$, moduli spaces of perverse coherent systems on the
resolved conifold, and a quiver which defines higher $\mathfrak{sl}_2$-spin
chains. Bethe equations are calculated in these cases.
  The construction in the present paper is based on the theory of gauged linear
sigma models as well as shifted symplectic geometry of Pantev, To\"en, Vaquie
and Vezzosi, and uses the virtual pullback formalism of symmetric obstruction
theory of Park, which arises from the recent development of Donaldson-Thomas
theory of Calabi-Yau 4-folds.
