Pseudoinstantons in metric-affine field theory

TL;DR

This paper introduces pseudoinstantons in metric-affine field theory as Lorentzian analogues of instantons, showing they are stationary points of a generalized Yang-Mills action and include Ricci-flat spacetimes and torsion waves.

Contribution

It develops a Lorentzian instanton construction in metric-affine theory, extending the concept of instantons beyond Euclidean space and linking it to Einstein vacuum solutions.

Findings

Ricci-flat Levi-Civita connections are pseudoinstantons.

A torsion wave solution is found in Minkowski space.

The theory includes vacuum Einstein equations as special cases.

Abstract

In abstract Yang-Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible ("pseudoinstanton"). We suggest a metric-affine action which is a natural generalization of the Yang-Mills action and for which pseudoinstantons are stationary points. We show that a spacetime with a Ricci flat Levi-Civita connection is a pseudoinstanton, so the vacuum Einstein equation is a special case of our theory. We also find another pseudoinstanton which is a wave of torsion in Minkowski space. Analysis of the latter solution indicates the possibility of using it as a model for the neutrino.