Homotopic Classification of Yang-Mills Vacua Taking into Account Causality

TL;DR

This paper investigates the classification of Yang-Mills vacuum states in asymptotically flat spacetimes, considering causality, and demonstrates that theta-vacua remain meaningful with a complexity similar to flat spacetime cases.

Contribution

It provides a homotopic classification of Yang-Mills vacua accounting for causality, showing theta-vacua are still relevant and comparable to flat spacetime scenarios.

Findings

Causality can make vacuum states homotopically equivalent in some cases.

Certain twisted classical vacuum states persist despite causality considerations.

The complexity of Yang-Mills vacua is similar to the flat Minkowskian case.

Abstract

Existence of theta-vacuum states in Yang--Mills theories defined over asymptotically flat space-times examined taking into account not only the topology but the complicated causal structure of these space-times, too. By a result of Galloway apparently causality makes all vacuum states, seen by a distant observer, homotopically equivalent making the introduction of theta-terms unnecessary. But a more careful analysis shows that certain twisted classical vacuum states survive even in this case eventually leading to the conclusion that the concept of ``theta-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. We also construct the theta vacua rigorously via geometric quantization.