Generalized Abelian Turaev–Viro and 𝑈(1) BF theories

Abstract

We explain how it is possible to study U ⁡ ( 1 ) \operatorname {U}(1) BF theory over a connected closed oriented smooth 3 3 -manifold in the formalism of path integral thanks to Deligne–Beilinson cohomology. We show how we can straightforwardly extend the definition to families of theories in any dimension. We then extend the definition of the Turaev–Viro invariant of a connected closed oriented smooth 3 3 -manifold in an Abelian framework to a family of invariants in any dimension. We show that those invariants can be written as discrete BF theories. We explain how the extensions of the U ⁡ ( 1 ) \operatorname {U}(1) BF theory we defined can be related to the extensions of the Turaev–Viro invariant we constructed.