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We explain how it is possible to study $\mathrm{U}\!\left(1\right)$ BF theoryover a connected closed oriented smooth $3$-manifold in the formalism of pathintegral thanks to Deligne-Beilinson cohomology. We show how we canstraightforwardly extend the definition to families of theories in anydimension. We extend then the definition of the Turaev-Viro invariant of aconnected closed oriented smooth $3$-manifold in an Abelian framework to afamily of invariants in any dimension. We show that those invariants can bewritten as discrete BF theories. We explain how the extensions of$\mathrm{U}\!\left(1\right)$ BF theory we defined can be related to theextensions of Turaev-Viro invariant we constructed.

Generalized Abelian Turaev-Viro and $\mathrm{U}\!\left(1\right)$ BF Theories