Nahm Poles and 0-Instantons

TL;DR

This paper investigates self-dual 0-connections on asymptotically hyperbolic 4-manifolds, revealing their asymptotic behavior, associated invariants, and relations to conformal geometry and energy functionals.

Contribution

It introduces the concept of 0-instanton obstruction tensor, proves its conformal invariance, and links the renormalized Yang-Mills energy to the Chern-Simons invariant.

Findings

The asymptotic expansion of 0-instantons is log-smooth.

The 0-instanton obstruction tensor is a conformal invariant.

The renormalized Yang-Mills energy equals the negative Chern-Simons invariant.

Abstract

We study self-dual 0-connections, or 0-instantons, on asymptotically hyperbolic 4-manifolds. These connections develop a uniform singularity along the conformal infinity, and are asymptotic, at each point of the boundary, to a "Nahm pole" model solution on $H^4$. Examples include the Levi-Civita spin connections on $S^+$ over spin Poincar\'e-Einstein 4-manifolds. Inspired by the Fefferman-Graham expansion for Poincar\'e-Einstein metrics, we study the asymptotic expansion of these 0-instantons. We prove that the expansion is log-smooth, and that the coefficient of the first log term - which we call the 0-instanton obstruction tensor - is a conformal invariant related to the Weyl curvature of the ambient conformal metric. We then show that this invariant vanishes if and only if the 0-instanton is smooth modulo gauge. Finally, we study the renormalized Yang-Mills energy: we prove that, if the metric is asymptotically Poincar\'e-Einstein to third order, then this energy is a well-defined conformal invariant, and equals the negative Chern-Simons invariant of the conformal infinity.