Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants

In this paper we identify the problem of equivariant vortex counting in a $(2,2)$ supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov-Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the ${\cal I}$ and ${\cal J}$-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov-Witten theory follow just by deforming the integration contour. In particular we apply our formalism to compute Gromov-Witten invariants of the $\mathbb{C}^3/\mathbb{Z}_n$ orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on $\mathbb {C}^2$ and of $A_n$ and $D_n$ singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.