Self-Excited Gravitational Instantons

TL;DR

The paper introduces self-excited gravitational instantons within the teleparallel formulation of general relativity, showing that the action can be written as a product of torsion and excitation forms and that solutions with $T^a=\pm H^a$ automatically satisfy the field equations. This leads to an action governed by the Nieh–Yan term, connecting axial torsion to a topological charge and recasting the gravitational action as a topological quantity. The Eguchi–Hanson instanton is shown to be a self-excited solution under appropriate gauge choices, with the Nieh–Yan charge computed to illustrate nontrivial topological sectors; other examples yield $\mathcal{N}=12\pi^2$, independent of certain function choices. Overall, the work provides a BPST-like topological perspective on gravitational instantons in teleparallel gravity and highlights the axial torsion as a topological current, offering a pathway to deeper understanding of the topological structure of GR.

Abstract

We present a novel approach to constructing gravitational instantons based on the observation that the gravitational action of general relativity in its teleparallel formulation can be expressed as a product of the torsion and excitation forms. We introduce a new class of solutions where these two forms are equal, which we term the self-excited instantons, and advocate for their use over the self-dual instantons of Eguchi and Hanson. These new self-excited instantons exhibit striking similarities to BPST instantons in Yang-Mills theory, as their action reduces to a topological Nieh-Yan term, which allows us to identify the axial torsion as a topological current and show that the gravitational action is given by a topological charge.