# Energy of sections of the Deligne-Hitchin twistor space

## TL;DR

This paper introduces a new energy functional on holomorphic sections of the Deligne-Hitchin space, linking it to harmonic maps, meromorphic connections, and Willmore energy, and uses it to identify new components.

## Contribution

It defines a novel energy functional on sections of the Deligne-Hitchin moduli space and connects it to existing geometric structures and energies, revealing new components.

## Key findings

- The energy functional generalizes the energy of equivariant harmonic maps.
- It relates to a meromorphic connection on a hyperholomorphic line bundle.
- The functional distinguishes new components of holomorphic sections.

## Abstract

We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines.