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

TL;DR

This paper explores a topological field theory involving holomorphic maps from complex 2-manifolds to toric targets, examining its mirror dual which involves free theories and topological strings, revealing new insights into string interactions.

Contribution

It introduces a novel model of holomorphic maps to toric manifolds, generalizing the A-model, and analyzes its mirror dual involving free theories and topological strings.

Findings

The model localizes to holomorphic maps with observables on (1,1)-submanifolds.

The mirror dual involves a free theory coupled with topological strings.

Vortex strings in the original model correspond to holomortex strings in the mirror.

Abstract

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 $(\mathbb{C}^*)^N$ 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 $N_\mathrm{comp}$ topological strings of type A. Here $N_\mathrm{comp}$ is the number of compactifying divisors of the toric target. Before the mirror transformation these strings are vortex (actually, holomortex) strings.