We consider the 4d compact U (1) gauge theory with extended actionWe give a full characterization of the phase diagram of this model using the notion of flux. The relation with the usual monopole picture is discussed. In analogy with the XY model we consider the helicity modulus [1] for this theory, and show that it is an order parameter. Analyzing the finite-size effects of the helicity modulus we conclude that the transition is first-order. The value of this order parameter is related to the renormalized coupling β R . We measure β c R at the transition point and give a counterexample to its conjectured universal value [2].

We consider the 4d compact U (1) theory with Wilson action and characterize its phase diagram using the notion of electromagnetic flux, instead of the more usual magnetic monopole. Taking inspiration from the flux picture, we consider the helicity modulus (h.m.) for this theory, and show that it is an order parameter for the confinement deconfinement phase transition. We extend the definition of the h.m. to an Abelian projected Yang-Mills theory, and discuss its behavior in SU (2).

We propose a new algorithm to compute the order-order interface tension in SU(N) lattice gauge theories. The algorithm is trivially generalizable to a variety of models, e.g., spin models. In the case N = 3, via the perfect wetting hypothesis, we can estimate the order-disorder interface tension. In the case N = 4, we study the ratio of dual k−tensions and find that it satisfies Casimir scaling down to T = 1.2 Tc.

If SU(N) gauge fields live in a world with a circular extra dimension, coupling there only to adjointly charged matter, the system possesses a global Z(N) symmetry. If the radius is small enough such that dimensional reduction takes place, this symmetry is spontaneously broken. It turns out that its fate at high temperatures is not easily decided with straightforward perturbation theory. Utilising non-perturbative lattice simulations, we demonstrate here that the symmetry does get restored at a certain temperature T c , both for a 3+1 and a 4+1 dimensional world (the latter with a finite cutoff). To avoid a cosmological domain wall problem, such models would thus be allowed only if the reheating temperature after inflation is below T c . We also comment on the robustness of this phenomenon with respect to small modifications of the model.

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