Instantons on Sasakian 7-manifolds

Abstract

AbstractWe study a natural contact instanton equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related to both the transverse Hermitian Yang–Mills (HYM) condition and the G2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreducible solutions. Following the approach by Baraglia and Hekmati in five dimensions [1], we derive cohomological conditions for smoothness and express its dimension in terms of the index of a transverse elliptic operator. Finally, we show that the moduli space of self-dual contact instantons is Kähler, in the Sasakian case. As an instance of concrete interest, we specialize to transversely holomorphic Sasakian bundles over contact Calabi–Yau 7-manifolds, as studied by Calvo-Andrade, Rodríguez and Sá Earp [8], and we show that in this context the notions of contact instanton, integrable G2-instanton and HYM connection coincide.