The Configuration Space Integral for Links and Tangles in R

The perturbative expression of Chern-Simons theory for links in Euclidean 3-space is a linear combination of integrals on configuration spaces. This has successively been studied by Guadagnini, Martellini and Mintchev, Bar-Natan, Kontsevich, Bott and Taubes, D. Thurston, Altschuler and Freidel, Yang. . .We give a self-contained version of this study with a new choice of compactification, and we formulate a rationality result. Then we generalise the construction to braids and tangles by constructing the suitable limit of the configuration spaces when we stretch the braid or tangle to a vertical line. We show that the limit integral is multiplicative and we define the corresponding connection. We also define a braiding and an associator which determine our limit for tangles. The braiding is given as a function of the Bott and Taubes anomaly. If the anomaly was zero in degrees > 4, then our limit integral for links would be equal to the Kontsevich integral.