On the energy spectrum of Yang--Mills instantons over asymptotically locally flat spaces

TL;DR

This paper proves that the energy of admissible SU(2) Yang--Mills instantons over ALF spaces is always an integer, extending previous results and suggesting a link to 2D conformal field theories.

Contribution

It establishes the integer energy property for Yang--Mills instantons over ALF spaces for SU(2) and U(2), refining earlier energy identities and proposing a connection to conformal field theory.

Findings

Energy of admissible SU(2) Yang--Mills instantons is always integer over ALF spaces.

The result extends to the gauge group U(2).

Potential link between 4D Yang--Mills on ALF spaces and 2D conformal field theory.

Abstract

In this paper we prove that over an asymptotically locally flat (ALF) Riemannian four-manifold the energy of an "admissible" SU(2) Yang--Mills is always integer. This result sharpens the previously known energy identity for such Yang--Mills instantons over ALF geometries. Furthermore we demonstrate that this statement continues to hold for the larger gauge group U(2). Finally we make the observation that there might be a natural relationship between 4 dimensional Yang--Mills theory over an ALF space and 2 dimensional conformal field theory. This would provide a further support for the existence of a similar correspondence investigated by several authors recently.