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Abelian duality is realized naturally by combining differential cohomologyand locally covariant quantum field theory. This leads to a C$^*$-algebra ofobservables, which encompasses the simultaneous discretization of both magneticand electric fluxes. We discuss the assignment of physically well-behavedstates to such algebra and the properties of the associated GNS triple. We showthat the algebra of observables factorizes as a suitable tensor product ofthree C$^*$-algebras: the first factor encodes dynamical information, while theother two capture topological data corresponding to electric and magneticfluxes. On the former factor we exhibit a state whose two-point correlationfunction has the same singular structure of a Hadamard state. Specifyingsuitable counterparts also on the topological factors we obtain a state for thefull theory, providing ultimately a unitary implementation of Abelian duality.

Hadamard states for quantum Abelian duality