A foliated Hitchin-Kobayashi correspondence

TL;DR

This paper extends the Hitchin-Kobayashi correspondence to foliated manifolds with transverse Hermitian structures, establishing stability conditions for foliated bundles and exploring their gauge-theoretic properties.

Contribution

It introduces a stability criterion for foliated Hermitian bundles and proves the existence of Hermitian-Einstein connections in this setting, adapting classical gauge theory results to foliations.

Findings

Hitchin-Kobayashi correspondence holds on compact Sasakian manifolds.

Foliated Hermitian-Einstein connections exist if and only if bundles are polystable.

Weak Uhlenbeck compactness fails for basic connections in foliated bundles.

Abstract

We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian-Einstein equations to higher dimensional instanton equations and in particular we look at the relation to higher contact instantons on Sasaki manifolds. For foliations of complex codimension 1, we obtain a transverse Narasimhan-Seshadri theorem. We also demonstrate that the weak Uhlenbeck compactness theorem fails in general for basic connections on a foliated bundle. This shows that not every result in gauge theory carries over to the foliated setting.