$\mathrm{Sp}(1)$-symmetric hyperkähler quantisation

Jørgen Ellegaard Andersen, Alessandro Malusà and Gabriele Rembado

Vol. 329 (2024), No. 1, 1–38

DOI: 10.2140/pjm.2024.329.1

Abstract

We provide a new general scheme for the geometric quantisation of $\mathrm{Sp}<!--FUNCTION\; APPLICATION-->(1)$-symmetric hyperkähler manifolds, considering Hilbert spaces of holomorphic sections with respect to the complex structures in the hyperkähler 2-sphere. Under properness of an associated moment map, or other finiteness assumptions, we construct unitary (super) representations of groups acting by Riemannian isometries preserving the 2-sphere, and we study their decomposition in irreducible components. We apply this scheme to hyperkähler vector spaces, the Taub–NUT metric on ${ℝ}^4$, moduli spaces of framed $\mathrm{SU}<!--FUNCTION\; APPLICATION-->(r)$-instantons on ${ℝ}^4$, and in part to the Atiyah–Hitchin manifold of magnetic monopoles in ${ℝ}^3$.

Keywords

geometric quantisation, hyperkähler geometry, moduli spaces, Hitchin connection, quantum representations

Mathematical Subject Classification

Primary: 53D50

Milestones

Received: 19 April 2022

Revised: 31 October 2023

Accepted: 15 March 2024

Published: 12 June 2024