Confining Membranes and Dimensional Reduction

TL;DR

This paper explores a dual theory of confining membranes and demonstrates how, under certain limits, the system reduces to a 2D sine-Gordon model, revealing insights into phase transitions and dimensional reduction in gauge theories.

Contribution

It constructs a generalized confining membrane theory and analyzes its dimensional reduction to a 2D integrable model, connecting 4D Coulomb gases with 2D sine-Gordon theory.

Findings

Dimensional reduction occurs in the strong coupling limit.

The reduced model is described by the 2D sine-Gordon theory.

Phase transition is of the Berezinskii-Kosterlitz-Thouless type.

Abstract

The dual theory describing the 4D Coulomb gas of point-like magnetically charged objects, which confines closed electric strings, is considered. The respective generalization of the theory of confining strings to confining membranes is further constructed. The same is done for the analogous SU(3)-inspired model. We then consider a combined model which confines both electric charges and closed strings. Such a model is nothing, but the mixture of the above-mentioned Coulomb gas with the condensate of the dual Higgs field, described by the dual Abelian Higgs model. It is demonstrated that in a certain limit of this dual Abelian Higgs model, the system under study undergoes naively the dimensional reduction and becomes described by the (completely integrable) 2D sine-Gordon theory. In particular, owing to this fact, the phase transition in such a model must be of the Berezinskii-Kosterlitz-Thouless type, and the respective critical temperature is expressed in terms of the parameters of the dual Abelian Higgs model. However, it is finally discussed that the dimensional reduction is rigorously valid only in the strong coupling limit of the original 4D Coulomb gas. In such a limit, this reduction transforms the combined model to the 2D free bosonic theory.