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

Abstract

We provide a new general scheme for the geometric quantisation of $\mathrm{Sp}(1)$-symmetric hyper-Kähler manifolds, considering Hilbert spaces of holomorphic sections with respect to the complex structures in the hyper-Kähler 2-sphere. Under properness of an associated moment map, or other finiteness assumptions, we construct unitary quantum (super) representations of central extensions of certain subgroups of Riemannian isometries preserving the 2-sphere, and we study their decomposition in irreducible components. We apply this quantisation scheme to hyper-Kähler vector spaces, the Taub--NUT metric on ${\mathbb{R}}^4$, moduli spaces of framed $\mathrm{SU}(r)$-instantons on ${\mathbb{R}}^4$, and partly to the Atiyah--Hitchin manifold of magnetic monopoles in ${\mathbb{R}}^3$

26 pages (plus references)

References in corpus (3)

• Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks
• Genus-one complex quantum Chern--Simons theory
• Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties