Séminaire Physique mathématique ICJ

Extensions of the Abelian Turaev-Viro construction and U(1) BF theory to any finite dimensional smooth oriented closed manifold

by Philippe Mathieu (Uni Zurich)

Fokko du Cloux (Bâtiment Braconnier)

Fokko du Cloux (Bâtiment Braconnier)

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.