Topological Dirac variables in Abelian $U(1)$ theory

In this study we, remembering the experience with topological Dirac variables in the non-Abelian Yang-Mills-Higgs (YMH) model with vacuuum BPS monopole solutions, attempt to construct similar for the Abelian $U(1)$ model. We show that QED, as one understands it commonly, is only the topologically trivial sector ($n=0$) of this Abelian $U(1)$ model. For $n\mathrm{\ne }0$ one gets Dirac monopole modes. In both the cases, $n=0$ and $n\mathrm{\ne }0$, the theory can be quantized via the Hamiltonian reduction in terms of Dirac variables.