Department of Mathematical Sciences,

University of Bath, Bath BA2 7AY, UK

E-mail: D.Vassiliev@bath.ac.uk

**The pseudoinstanton - a Lorentzian analogue of the
instanton**

**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 (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. In the final part of the talk we present examples of pseudoinstantons
and discuss them within the context of non-Riemannian theories of gravity.
The talk is an extension of the author's paper [1].

**References:**

- D. Vassiliev, Pseudoinstantons in metric-affine field theory, General Relativity and Gravitation, 2002, vol. 34, p. 1239-1265.