The local moduli space of the Einstein-Yang-Mills system

TL;DR

This paper investigates the local structure of the moduli space of Einstein-Yang-Mills pairs on compact manifolds, extending classical results and introducing the concept of essential deformations, with special focus on four-dimensional cases.

Contribution

It constructs a smooth slice for the Einstein-Yang-Mills system, characterizes the moduli space as an analytic set, and introduces essential deformations with detailed analysis in four dimensions.

Findings

Realized the moduli space as an analytic set in a tame Fréchet manifold.

Introduced and characterized the notion of essential deformation.

Proved restrictions on essential deformations in four-dimensional Calabi-Yau cases.

Abstract

We study the deformation theory of the Einstein-Yang-Mills system on a principal bundle with a compact structure group over a compact manifold. We first construct, as an application of the general slice theorem of Diez and Rudolph, a smooth slice in the tame Fr\'echet category for the coupled action of bundle automorphisms on metrics and connections. Using this result, together with a careful analysis of the linearization of the Einstein-Yang-Mills system, we realize the moduli space of Einstein-Yang-Mills pairs modulo automorphism as an analytic set in a finite-dimensional tame Fr\'echet manifold, extending classical results of Koiso for Einstein metrics and Yang-Mills connections to the Einstein-Yang-Mills system. Furthermore, we introduce the notion of \emph{essential deformation} of an Einstein-Yang-Mills pair, which we characterize in full generality and explore in more detail in the four-dimensional case, proving a decoupling result for trace deformations when the underlying Einstein-Yang-Mills pair is a Ricci-flat metric coupled to an anti-self-dual instanton. In particular, we find a novel obstruction that does not occur in the \emph{decoupled} Einstein or Yang-Mills moduli problems. Finally, we prove that every essential deformation of the four-dimensional Einstein-Yang-Mills system based on a Calabi-Yau metric coupled to an instanton is of restricted type.