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Gauge Theory on Projective Surfaces and Anti-self-dual Einstein Metrics in Dimension Four.
The Journal of Geometric Analysis ( IF 1.1 ) Pub Date : 2017-10-12 , DOI: 10.1007/s12220-017-9934-9
Maciej Dunajski 1 , Thomas Mettler 2
Affiliation  

Given a projective structure on a surface \(N\), 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 \(M \rightarrow N\). 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 \(\mathbb {RP}^2\) is the non-compact real form of the Fubini–Study metric on \(M=\mathrm {SL}(3, \mathbb {R})/\mathrm {GL}(2, \mathbb {R})\). We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.

中文翻译:

投影曲面的量规理论和第四维反自对偶爱因斯坦度量标准。

给定的表面上的投影结构\(N \) ,我们将展示如何规范地构建一个中性签名爱因斯坦度量具有非零数量曲率以及总空间中的辛形式中号一定秩2仿射束\( M \ rightarrow N \)。爱因斯坦度量具有反自对偶共形曲率,并允许反自对偶平面的平行场。我们显示出,本地每个此类度量均来自我们的构造,除非它是共形的。与\(\ mathbb {RP} ^ 2 \)上的平面投影结构相对应的齐次爱因斯坦度量是\(M = \ mathrm {SL}(3,\ mathbb上)的Fubini–Study度量的非紧实形式。{R})/ \ mathrm {GL}(2,\ mathbb {R})\)。我们还展示了我们的构造与Calderbank引入的某个规范理论方程之间的关系。
更新日期:2017-10-12
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