$\operatorname{Sp}(1)$-symmetric hyperkähler quantisation
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910492374597632 |
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| author | Andersen, Jørgen Ellegaard Malusà, Alessandro Rembado, Gabriele |
| author_facet | Andersen, Jørgen Ellegaard Malusà, Alessandro Rembado, Gabriele |
| contents | 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 $\mathbb{R}^3$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_03584 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $\operatorname{Sp}(1)$-symmetric hyperkähler quantisation Andersen, Jørgen Ellegaard Malusà, Alessandro Rembado, Gabriele Differential Geometry Mathematical Physics 53C26 53D50 (primary), 14D21 (secondary) 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 $\mathbb{R}^3$ |
| title | $\operatorname{Sp}(1)$-symmetric hyperkähler quantisation |
| topic | Differential Geometry Mathematical Physics 53C26 53D50 (primary), 14D21 (secondary) |
| url | https://arxiv.org/abs/2111.03584 |