Topological Flux Sectors in Extended U(1) Gauge Theory on T

We consider the 4d compact U (1) gauge theory with fundamental-adjoint action on a hypertorus. We give a full characterization of the phase diagram of this model in terms of topological flux sectors. 1. The flux in abelian gauge theories Consider an abelian gauge theory defined on a hypercube of size L and periodic boundary conditions. In the continuum the definition of flux through any (µ, ν) plane is the following: Φ µν = F µν dσ (1) This quantity, because of periodic boundary conditions , is 2πk valued (k ∈ Z), and configurations with different k values are topologically disconnected (so we talk of topological superselection sectors). Consider now the same system, but with a dis-cretized space-time. The definition of flux is now Φ µν = 1 L 2 µν planes P [θ Pµν ] −π,π (2) where [θ P ] −π,π is the plaquette angle reduced to the interval (−π, π), its so called physical part. A double sum is present: the internal P is the sum over the plaquettes in a single (µ, ν) plane; this quantity is a multiple of 2π, as in the continuum case. But when we consider the external average over all parallel planes, we observe that the flux can change from plane to plane, due to the presence of magnetic monopoles, specific to the lattice, so that the allowed values for Φ µν are multiples of (2π)/L 2. Having in mind the continuum limit, configurations whose flux is 2πk play a special role: in fact