Twisted hyperk\"ahler symmetries and hyperholomorphic line bundles

TL;DR

This paper introduces new concepts of non-isometric symmetries in hyperk"ahler spaces, linking them to hyperholomorphic line bundles and exploring their implications in various geometric and physical contexts.

Contribution

It defines and studies new types of hyperk"ahler symmetries, expanding the understanding of hyperk"ahler geometry and its connections to line bundles and twistor theory.

Findings

Hyperk"ahler symmetries decompose into tri-Hamiltonian and rotational types.

Hyperholomorphic line bundles are associated with rotational symmetries.

Applications include c-map metrics and hyperk"ahler structures in integrable systems.

Abstract

In this paper we propose and investigate in full generality new notions of (continuous, non-isometric) symmetry on hyperk\"ahler spaces. These can be grouped into two categories, corresponding to the two basic types of continuous hyperk\"ahler isometries which they deform: tri-Hamiltonian isometries, on one hand, and rotational isometries, on the other. The first category of deformations gives rise to Killing spinors and generate what are known as hidden hyperk\"ahler symmetries. The second category gives rise to hyperholomorphic line bundles over the hyperk\"ahler manifolds on which they are defined and, by way of the Atiyah-Ward correspondence, to holomorphic line bundles over their twistor spaces endowed with meromorphic connections, generalizing similar structures found in the purely rotational case by Haydys and Hitchin. Examples of hyperk\"ahler metrics with this type of symmetry include the c-map metrics on cotangent bundles of affine special K\"ahler manifolds with generic prepotential function, and the hyperk\"ahler constructions on the total spaces of certain integrable systems proposed by Gaiotto, Moore and Neitzke in connection with the wall-crossing formulas of Kontsevich and Soibelman, to which our investigations add a new layer of geometric understanding.