Homotopic Classification of Yang–Mills Vacua Taking into Account Causality

In this letter we study the existence of θ-vacuum states in Yang–Mills theories defined over asymptotically flat, stationary space-times taking into account not only the topology but the complicated causal structure of these space-times, too. By a result of Chruściel and Wald, apparently causality makes all vacuum states, seen by a distant observer, homotopically equivalent making the introduction of θ-terms unnecessary in (causally) effective Lagrangians. But a more careful study shows that certain twisted classical vacuum states survive even in this case eventually leading to the conclusion that the concept of “θ-vacua” is meaningful in the case of general Yang–Mills theories. We give a classification of these vacuum states based on Isham’s results showing that the Yang–Mills vacuum has the same complexity as in the flat Minkowskian case hence the general CP-problem is not more complicated than the well-known flat one.