The continuum approach to the BF vacuum: the U(1) case

A quantum representation of holonomies and exponentiated fluxes of a $U(1)$ gauge theory that contains the Pullin-Dittrich-Geiller (DG) vacuum is presented and discussed. Our quantization is performed manifestly in a continuum theory, without any discretization. The discretness emerges on the quantum level as a property of the spectrum of the quantum holonomy operators. The new type of a cylindrical consistency present in the DG approach, now follows easily and naturally. A generalization to the non--Abelian case seems possible.