Geometric Interpretation of Schwarzschild Instantons

TL;DR

This paper constructs and analyzes non-topological Abelian instantons with finite energy on the Euclidean Schwarzschild manifold, linking them to known SU(2)-instantons and calculating the harmonic space structure.

Contribution

It introduces a new non-topological L^2 harmonic 2-form on the Euclidean Schwarzschild manifold and relates it to existing SU(2)-instantons, expanding understanding of instanton solutions.

Findings

Found a non-topological self-dual L^2 harmonic 2-form on the Euclidean Schwarzschild manifold.

Identified these forms with SU(2)-instantons of Pontryagin number 2n^2.

Calculated the full L^2 harmonic space for the manifold.

Abstract

In this note we address the problem of finding Abelian instantons of finite energy on the Euclidean Schwarzschild manifold. This amounts to construct self-dual L^2 harmonic 2-forms on the space. Gibbons found a non-topological L^2 harmonic form in the Taub-NUT metric, leading to Abelian instantons with continuous energy. We imitate his construction in the case of the Euclidean Schwarzschild manifold and find a non-topological self-dual L^2 harmonic 2-form on it. We show how this gives rise to Abelian instantons and identify them with SU(2)-instantons of Pontryagin number 2n^2 found by Charap and Duff in 1977. Using results of Dodziuk and Hitchin we also calculate the full L^2 harmonic space for the Euclidean Schwarzschild manifold.