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Gauge Theory on Projective Surfaces and Anti-self-dual Einstein Metrics in Dimension Four.

ABSTRACT: Given a projective structure on a surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}N, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space M of a certain rank 2 affine bundle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \rightarrow N$$\end{document}M?N. The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {RP}^2$$\end{document}RP2 is the non-compact real form of the Fubini–Study metric on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=\mathrm {SL}(3, \mathbb {R})/\mathrm {GL}(2, \mathbb {R})$$\end{document}M=SL(3,R)/GL(2,R). We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.

SUBMITTER: Dunajski M

PROVIDER: S-EPMC6294184 | biostudies-other | 2018

REPOSITORIES: biostudies-other

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Gauge Theory on Projective Surfaces and Anti-self-dual Einstein Metrics in Dimension Four.

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