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

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.

On the global structure of deformed Yang-Mills theory and QCD(adj) on ℝ 3 × S 1 $$ {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}^1 $$

“…For q > 0, we notice that Pw w w k + w w w q = w w w k+q(modNc) (mod Γ r ). This implies that all N c minima within Γ r are identified under the action ofγ (1,q) with q > 0 and thus the [SU(N c )/Z Nc ] q>0 theories have unique ground states, as shown on the right panel of figure 5 for SU (3).…”

“…Lattice gauge theories for different choice of gauge group with the same algebra have also been the studied, see e.g. [3] and references therein. Interestingly, it was only recently realized that even when the gauge group is chosen, there is a further set of discrete parameters, called "discrete θ-angles" in [4], that label different theories with the same gauge group (we refer to the choice of gauge group and discrete θ-angle parameters as "global structure").…”

“…To the best of our knowledge, the Kramers-Wannier duality of the effective theory is the only example of an electric-magnetic duality in the framework of nonsupersymmetric pure YM theory. 3 …”

“…Figure 2. dYM : the σ σ σ 2π plane for su (3). The SU(3) fundamental domain is Γ w , spanned by w w w 1,2 .…”

't Hooft loop and the phases of SU(2) lattice gauge theory

“…The natural choice to investigate F upon lattice discretization of Yang-Mills theories would be to define the partition function through the adjoint Wilson action Z ∼ e β A Tr A (U) ; in this case all topological sectors are dynamically included [8]. Universality should of course allow the equivalent use of the standard Wilson plaquette action S ∼ Tr F (U).…”