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arXiv · 2103.01549

Anti-self-dual connections over the $5$D Heisenberg group and the twistor method

Guangzhen Ren·Wei Wang
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Abstract

In this paper, we introduce notions of $\alpha$-planes in $5$D complex Heisenberg group and the twistor space as the moduli space of all $\alpha$-planes. So we can define an anti-self-dual (ASD) connection as a connection flat over all $\alpha$-planes. This geometric approach allows us to establish Penrose-Ward correspondence between ASD connections over $5$D complex Heisenberg group and a class of holomorphic vector bundles on the twistor space. By Atiyah-Ward ans\"{a}tz, we also construct a family of ASD connections on $5$D complex Heisenberg group. When restricted to $5$D real Heisenberg group, the flat model of $5$D contact manifolds, an ASD connection satisfies the horizontal part of the contact instanton equation introduced by physicists.

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Guangzhen Ren, Wei Wang. 2021-03-02. Anti-self-dual connections over the $5$D Heisenberg group and the twistor method. https://arxiv.org/abs/2103.01549

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