Comparison of SO(3) and SU(2) lattice gauge theory

TL;DR

This paper compares SO(3) and SU(2) lattice gauge theories, analyzing their topological features, phase transitions, and the role of the gauge group's center, with implications for understanding confinement.

Contribution

It introduces lattice observables for topological invariants in SO(3), compares phase behavior with SU(2), and discusses an analytic connection at weak coupling.

Findings

SO(3) phase with negative adjoint Polyakov loop explained by topological observables

Electric twist free energy used to calibrate the confinement-deconfinement transition

Large lattices needed to fully study the SO(3) confined phase

Abstract

The Villain form of SO(3) lattice gauge theory is studied and compared to Wilson's SU(2) theory. The topological invariants in SO(3) which correspond to twisted boundary conditions in SU(2) are discussed and lattice observables are introduced for them. An apparent SO(3) phase with negative adjoint Polyakov loop is explained in terms of these observables. The electric twist free energy, an order parameter for the confinement-deconfinement transition, is measured in both theories to calibrate the temperature. The results indicate that lattices with about 700^4 sites or larger will be needed to study the SO(3) confined phase. Alternative actions are discussed and an analytic path connecting SO(3) and SU(2) lattice gauge theory at weak coupling is exhibited. The relevance for confinement of the centre of the gauge group is discussed.