A New Gauge-Theoretic Construction of 4-Dimensional Hyperkähler ALE Spaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918276787863552 |
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| author | Yan, Jiajun |
| author_facet | Yan, Jiajun |
| contents | Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence.
The 4-dimensional hyperkähler ALE spaces are first classified by Peter Kronheimer via a finite-dimensional hyperkähler reduction. In this paper, we give a new gauge-theoretic construction of these spaces. More specifically, we realize each 4-dimensional hyperkähler ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a gauge group action. The construction given in this paper parallels Kronheimer's original construction and hence can also be thought of as a gauge-theoretic interpretation of Kronheimer's construction of these spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10979 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A New Gauge-Theoretic Construction of 4-Dimensional Hyperkähler ALE Spaces Yan, Jiajun Differential Geometry Symplectic Geometry 53D20, 53D30, 14J60, 32L05, 57M99 Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. The 4-dimensional hyperkähler ALE spaces are first classified by Peter Kronheimer via a finite-dimensional hyperkähler reduction. In this paper, we give a new gauge-theoretic construction of these spaces. More specifically, we realize each 4-dimensional hyperkähler ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a gauge group action. The construction given in this paper parallels Kronheimer's original construction and hence can also be thought of as a gauge-theoretic interpretation of Kronheimer's construction of these spaces. |
| title | A New Gauge-Theoretic Construction of 4-Dimensional Hyperkähler ALE Spaces |
| topic | Differential Geometry Symplectic Geometry 53D20, 53D30, 14J60, 32L05, 57M99 |
| url | https://arxiv.org/abs/2310.10979 |