Twisted hyperkähler symmetries and hyperholomorphic line bundles

Abstract

In this paper we propose and investigate in full generality new notions of (continuous, non-isometric) symmetry on hyperkähler spaces. These can be grouped into two categories, corresponding to the two basic types of continuous hyperkähler 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 hyperkähler symmetries. The second category gives rise to hyperholomorphic line bundles over the hyperkähler 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 hyperkähler metrics with this type of symmetry include the c-map metrics on cotangent bundles of affine special Kähler manifolds with generic prepotential function, and the hyperkähler 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.