$\mathrm{Sp}(1)$-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 ${\mathbb{R}}^3$

arXiv comment26 pages (plus references)

References in corpus (15)

• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• Instanton counting on blowup. I. 4-dimensional pure gauge theory
• Quasi-Hamiltonian Geometry of Meromorphic Connections
• Geometric construction of modular functors from Conformal Field Theory
• Abelian Conformal Field theories and Determinant Bundles
• Global Weyl groups and a new theory of multiplicative quiver varieties
• Berezin-Toeplitz quantization and Berezin transform
• Equivariant indices of Spin${}^c$-Dirac operators for proper moment maps
• Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks
• A fixed point theorem on noncompact manifolds
• Genus-one complex quantum Chern--Simons theory
• Stratified hyperkahler spaces from semisimple Lie algebras
• Kempf-Ness type theorems and Nahm equations
• Asymptotic properties of the Hitchin-Witten connection
• Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

Cited by in corpus (1)

• A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT