In 1992, V. Turaev and O. Viro defined an invariant of smooth oriented closed 3-manifolds consisting of labelling the edges of a triangulation of the manifold with representations of a deformation of the enveloping algebra of sl(2,C) at a root of unity, associating a (quantum) 6j-symbol to each tetrahedron of the triangulation, taking the product of the 6j-symbols over all the tetrahedra of the manifold, then summing over all the admissible labelling representations. It is commonly admitted that this construction is a regularization of a path integral occurring in quantum gravity, the so-called “Ponzano-Regge model”, which is a kind of SU(2) BF gauge theory. A naive question is: Is it possible to define an abelian version of this invariant? If yes, is there a relation with an abelian BF gauge theory? These questions were answered positively in 2016, and the corresponding Turaev-Viro invariant is built from Z/kZ labelling representations (the equivalent of 6j-symbols being ``modulo k'' Kronecker symbols) while the associated gauge theory is a particular U(1) BF theory (with coupling constant k). This U(1) BF theory can be straightforwardly extended to any finite dimensional closed oriented manifold, and so can be the Turaev-Viro construction built from Z/kZ labelling representations. A natural question is thus: Are these extensions still related? I will answer this question during the talk.
