Theory of holomorphic maps of two-dimensional complex manifolds to toric manifolds and type-a multi-string theory

We study the field theory localizing to holomorphic maps from a complex manifold of complex dimension $2$ to a toric target (a generalization of A model). Fields are realized as maps to ${{(\mathbb{C}\text{*})}^{N}}$ where one includes special observables supported on $(1,1)$-dimensional submanifolds to produce maps to the toric compactification. We study the mirror of this model. It turns out to be a free theory interacting with ${{N}_{{{\text{comp}}}}}$ topological strings of type A. Here, ${{N}_{{{\text{comp}}}}}$ is the number of compactifying divisors of the toric target. Before the mirror transformation, these strings are vortex (actually, holomortex) strings.