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

By introducing notions of an α-plane in the 5D complex Heisenberg group and the twistor space as the moduli space of all α-planes, we can define an anti-self-dual (ASD) connection as a connection flat over all α-planes. This geometric approach allows us to establish the Penrose-Ward correspondence between ASD connections over the 5D complex Heisenberg group and a class of holomorphic vector bundles on the twistor space. By Atiyah-Ward ansätz, we also construct ASD connections on the 5D complex Heisenberg group. When restricted to the 5D real Heisenberg group, the flat model of 5D contact manifolds, an ASD connection satisfies the horizontal part of the contact instanton equation.