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

Jørgen Ellegaard Andersen, Alessandro Malusà, Gabriele Rembado

Abstract

We provide a new general scheme for the geometric quantisation of $\operatorname{Sp}(1)$-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 $\operatorname{SU}(r)$-instantons on $\mathbb{R}^4$, and partly to the Atiyah--Hitchin manifold of magnetic monopoles in $\mathbb{R}^3$

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

Abstract

We provide a new general scheme for the geometric quantisation of -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 , moduli spaces of framed -instantons on , and partly to the Atiyah--Hitchin manifold of magnetic monopoles in

Paper Structure

This paper contains 23 sections, 20 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Description of the main construction
  3. Applications and further directions
  4. Abstract Sp(1)-symmetric hyperkähler quantisation
  5. Hyperkähler manifolds and their symmetry groups
  6. Pre-quantum data
  7. Geometric quantisation
  8. Decomposition of H-q
  9. Structure of H-lambda
  10. Quantum super Hilbert spaces and unitary representations
  11. Finite-rank conditions
  12. Rank-generating series and localisation formulæ
  13. Sp(1)-symmetric hyperkähler potentials
  14. Examples of applications
  15. Hyperkähler vector spaces
  16. ...and 8 more sections

Key Result

Theorem 1.1

Suppose that For each other symplectic form $\omega_{q'}$, call $\mathcal{H}_{q'}^{(\rho)}$ the isotypical component in $\mathcal{H}_{q'}$ corresponding to $\rho$ under the identification $G_{q} \simeq G_{q'}$ by conjugation in $G$. Then the collection of spaces $\mathcal{H}^{(\rho)}$ has a canonical structure a preserving the Hermitian structure and connection, where $\underline{V_{\rho}} = \mat

Theorems & Definitions (45)

  • Theorem 1.1: Cf. Thm \ref{['thm:CurvaturePF']}
  • Theorem 1.2: Cf. § \ref{['sec:super_spaces']}
  • Theorem 1.3: Cf. Thmm. \ref{['thm:fdmm']} and \ref{['thm:formulaHHprime']}
  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 35 more