Yang-Mills Connections on Conformally Compact Manifolds

TL;DR

This paper investigates the structure of Yang-Mills connections on conformally compact manifolds, proving existence and extension results for small boundary deformations, and confirming a conjecture related to holography.

Contribution

It establishes the existence of interior Yang-Mills connections extending small boundary deformations, under a nondegeneracy condition, confirming a holography-related conjecture.

Findings

Existence of interior Yang-Mills connections for small boundary deformations

Extension of boundary data to interior solutions under nondegeneracy

Confirmation of Witten's holography conjecture

Abstract

We study the moduli space of Yang--Mills connections on bundles over a conformally compact manifold $\overline{M}$. We prove that, for every Yang--Mills connection $A$ that satisfies an appropriate nondegeneracy condition, and for every small deformation $\gamma$ of $A_{|\partial\overline{M}}$, there is a Yang--Mills connection in the interior that extends $A_{|\partial\overline{M}}+\gamma$. As a corollary, we confirm an expectation of Witten mentioned in his foundational paper about holography [arXiv:hep-th/9802150].