Renormalized Yang-Mills Energy on Poincar\'e-Einstein Manifolds

TL;DR

This paper establishes a link between the renormalized Yang-Mills energy in six-dimensional Poincaré-Einstein manifolds and conformally invariant integrands, connecting bulk and boundary energies through advanced geometric analysis.

Contribution

It provides a new expression for the renormalized Yang-Mills energy as a conformally invariant bulk integral and relates it to boundary anomalies using generalized geometric methods.

Findings

Renormalized energy expressed as a conformally invariant bulk integral.

Boundary anomaly integrand matches the bulk integrand in seven dimensions.

Method extends Chang-Qing-Yang approach to higher-dimensional Poincaré-Einstein spaces.

Abstract

We prove that the renormalized Yang-Mills energy on six dimensional Poincar\'e-Einstein spaces can be expressed as the bulk integral of a local, pointwise conformally invariant integrand. We show that the latter agrees with the corresponding anomaly boundary integrand in the seven dimensional renormalized Yang-Mills energy. Our methods rely on a generalization of the Chang-Qing-Yang method for computing renormalized volumes of Poincar\'e-Einstein manifolds, as well as known scattering theory results for Schr\"odinger operators with short range potentials.