A first analysis of the ensemble of local maxima of maximal center gauge

TL;DR

This paper analyzes the ensemble of local maxima in maximal center gauge, revealing insights into vortex detection and the limitations of current maximization methods for understanding confinement in gauge theories.

Contribution

It provides the first analysis of local maxima in maximal center gauge and links their structure to vortex detection and string tension estimation.

Findings

String tension decreases linearly with local maxima gauge functional value.

Ensemble of local maxima contains information about vortex positions.

Global maximization of the gauge functional is practically unattainable.

Abstract

Maximal center gauge (MCG) aims to detect some of the most important vacuum configurations, suggesting thick magnetic flux tubes quantised to non-trivial center elements of the gauge group being responsible for confinement. Due to the NP-hardness of a global maximization of the gauge functional only numeric procedures aiming for local maxima are possible. We observe a linear decrease of the string tension with increasing gauge functional value of the local maxima. This implies that the request to get as close as possible to the absolute maximum is untainable. We compare global properties of the ensemble of local maxima with other methods to detect center vortices and with determinations of the string tension from full configurations. This comparison indicates that the information about the number and positions of center vortices is contained in the structure of the ensemble of local maxima. This may pave the way for a future more successful formulation of the gauge condition.