Vassiliev invariants for flows via ChernSimons perturbation theory
Abstract
The perturbative expansion of ChernSimons gauge theory leads to invariants of knots and links, the socalled finite type invariants or Vassiliev invariants. It has been proved that at any order in perturbation theory the superposition of certain amplitudes is an invariant of that order. BottTaubes integrals on configuration spaces are introduced in the present context to write Feynman diagrams at a given order in perturbation theory in a geometrical and topological framework. One of the consequences of this formalism is that the resulting amplitudes are rewritten in cohomological terms in configuration spaces. This cohomological structure can be used to translate BottTaubes integrals into ChernSimons perturbative amplitudes and vice versa. In this paper, this program is performed up to third order in the coupling constant. This expands some work previously worked out by Thurston. Finally we take advantage of these results to incorporate in the formalism a smooth and divergenceless vector field on the 3manifold. The BottTaubes integrals obtained are used for constructing higherorder average asymptotic Vassiliev invariants extending the work of Komendarczyk and Volić.
 Publication:

International Journal of Modern Physics A
 Pub Date:
 May 2021
 DOI:
 10.1142/S0217751X21500895
 arXiv:
 arXiv:2004.13893
 Bibcode:
 2021IJMPA..3650089D
 Keywords:

 Chern–Simons perturbation theory;
 Bott–Taubes integration;
 average asymptotic invariants;
 11.15.Yc;
 12.38.Bx;
 Perturbative calculations;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 54 pages, 21 figures, 3 appendices