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Philippe Mathieu: Extensions of the Abelian Turaev-Viro construction and U(1) BF theory to any finite dimensional smooth oriented closed manifolds.

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mercredi 23 mars 2022 mercredi 23 mars 2022

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 𝒰q(sl₂(ℂ)) (q being 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 during my PhD in 2016, and the corresponding Turaev-Viro invariant is built from ℤ/kℤ labelling representations (the equivalent of 6j-symbols being « mod 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 smooth oriented closed manifolds of any dimension, 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.


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