The Pseudoinstanton – a Lorentzian Analogue of the Instanton

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 (i.e. connection on vectors) an instanton-type construction can be carried out in the Lorentzian setting. Namely, the Lorentzian analogue of an instanton is a metric compatible connection whose curvature is irreducible and simple (“pseudoinstanton”). We prove that a pseudoinstanton is a solution of the Yang–Mills equation for the affine connection. In fact, we prove a much stronger result: a pseudoinstanton is a stationary point of any Lorentz-invariant quadratic action with respect to the independent variation of the metric and the connection. We present examples of pseudoinstantons and discuss them within the context of non-Riemannian theories of gravity.