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

TL;DR

This paper develops a geometric framework for anti-self-dual connections on the 5D Heisenberg group using twistor theory, establishing a correspondence with holomorphic bundles and constructing explicit ASD connections.

Contribution

It introduces the notion of -planes in 5D complex Heisenberg groups and links ASD connections to twistor space, extending Penrose-Ward correspondence to this setting.

Findings

Established Penrose-Ward correspondence for 5D Heisenberg group

Constructed explicit ASD connections using Atiyah-Ward ansatz

Connected ASD connections to contact instanton equations in 5D

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.