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Sur une anomalie du développement perturbatif de la théorie de Chern-Simons

Abstract : The Kontsevich invariant Z of rational homology 3- sphere was constructed by Maxim Kontsevich in 1992 using configuration space integrals.This invariant is graduated. It can be written as Z=(Zn){nin NN}, where Zn values in the space mathcal{A} n of jacobi diagram with order n. A Jacobi diagram with order n is a trivalent graph with 2n vertices. At a first point, we can see Z as an invariant Z(M,tau) of rational homology 3-spheres equipped with a trivialisation tau so that Z is the exponential of an invariant z(M,tau)=(zn(M,tau)) {ninNN}. In fact, we can say that zn(M,tau) counts the number of embeddings of connected jacobi diagrams with order n with some additionnal conditions. We can associate an homotopic integer invariant p 1(tau) to each trivialisation tau of oriented 3-manifolds and it exists beta=(betan){ninNN}, where betaninmathcal{A} n that is called anomaly so that zn(M,tau) - p 1(tay) is independant of tau. We name it zn(M) and Z(M)=exp((zn(M){nin NN})).Greg Kuperberg and Dylan Thurston introduced this constant in 1999. We already know that betan=0 if n is even and beta 1neq 0. This thesis is about the computation of beta 3. It describes simplified expressions of beta 3, and this expressions can be compute with a computer.
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Submitted on : Wednesday, January 10, 2018 - 10:05:06 AM
Last modification on : Tuesday, October 6, 2020 - 8:36:03 AM
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Kévin Corbineau. Sur une anomalie du développement perturbatif de la théorie de Chern-Simons. Topologie générale [math.GN]. Université Grenoble Alpes, 2016. Français. ⟨NNT : 2016GREAM038⟩. ⟨tel-01679640⟩



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