# J-holomorphic curves and Dirac-harmonic maps

## TL;DR

This paper explores the relationship between J-holomorphic curves and Dirac-harmonic maps, showing how certain maps and spinors form Dirac-harmonic maps on Riemann surfaces with implications for topological string theory.

## Contribution

It identifies conditions under which J-holomorphic curves on Riemann surfaces generate Dirac-harmonic maps into Kähler manifolds, linking geometric analysis with string theory models.

## Key findings

- Construction of Dirac-harmonic maps from J-holomorphic curves.
- Description of the moduli space tangent bundle as Dirac-harmonic maps.
- Connection to the A-model in topological string theory.

## Abstract

Dirac-harmonic maps are critical points of a fermionic action functional, generalizing the Dirichlet energy for harmonic maps. We consider the case where the source manifold is a closed Riemann surface with the canonical Spin^c-structure determined by the complex structure and the target space is a Kaehler manifold. If the underlying map f is a J-holomorphic curve, we determine a space of spinors on the Riemann surface which form Dirac-harmonic maps together with f. For suitable complex structures on the target manifold the tangent bundle to the moduli space of J-holomorphic curves consists of Dirac-harmonic maps. We also discuss the relation to the A-model of topological string theory.