Abstract
AbstractNon-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of complete 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.
Publisher
Springer Science and Business Media LLC
Reference22 articles.
1. Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 308(1505), 523–615 (1983)
2. Atiyah, M.F., Hitchin, N.J.: The Geometry and Dynamics of Magnetic Monopoles, vol. 8. Princeton University Press, Princeton (2014)
3. Boalch, P.: Simply-laced isomonodromy systems. Publ. Math. l’IHÉS 116, 1–68 (2012)
4. Cherkis, S.A., Kapustin, A.: Singular monopoles and gravitational instantons. Commun. Math. Phys. 203, 713–728 (1999)
5. Cherkis, S.A., Kapustin, A.: Singular monopoles and supersymmetric gauge theories in three dimensions. Nucl. Phys. B 525(1–2), 215–234 (1998)