The first non-vanishing Massey-Milnor linkings are constructed in a systematic way by summing Chern-Simons-Witten configuration space integrals. In Section 3 (Definition 3.1), to any oriented and ordered link \(L=\{L_0,\ldots,L_n\}\) in \(\mathbb{R}^3\) with base points \(x_j \in L_j\), the authors associate inductively a formal sum of specific instances of oriented and uni-trivalent rooted trees, called Chern-Simons-Witten graphs and defined in Section 2, where the univalent vertices are supposed to be lying on the \(L_j\). For any such a graph \(\Gamma\) there is a configuration space \(\mathcal{C}\Gamma\), a product of:
\(\bullet\) a copy of \(\mathbb{R}^3\) for each trivalent vertex of \(\Gamma\),
\(\bullet\) \(L_j\) for each univalent vertex lying on the component \(L_j\) of \(L\),
\(\bullet\) sets of the form \(\{(y_1,y_2,\dots,y_k)\mid x_j\leq y_1\leq y_2\leq \dots\leq y_k \leq x_j\}\subset (L_j)^k\) corresponding to linearly ordered \(k\)-tuples of univalent vertices of \(\gamma\) on \(L_j\).
The configuration space integral of \(\Gamma\) is the integral over \({\mathcal C}\Gamma\) of the differential form constructed by taking the product, indexed by all edges of \(\Gamma\), of the pull-back of the standard area form on the unit sphere \(S^2\) via the map \[ X_1 \times X_2\rightarrow S^2,\quad (A,B) \mapsto \frac{A-B}{\mid A-B\mid} \] where \(X_i={\mathbb R}^3\) or some \(L_j\) (Definition 2.1). The main result of the paper is the topological invariance of \(L_{m,m-1,\ldots,0}\), defined as the sum of configuration space integrals associated to the formal sum of Chern-Simons-Witten graphs for \(L=\{L_0,\ldots,L_n\}\), when the corresponding sum vanishes for any proper sublink of \(L\) (Theorem 4.2). This is proved in Section 4; the idea is to show that the variation of \(L_{m,m-1,\ldots,0}\), interpreted as a \(1\)-form on the space of links of \((n+1)\)-components, is zero. Connections of \(L_{m,m-1,\ldots,0}\) with the Massey-Milnor linkings are discussed in section 5. In particular, the invariance of \(L_{m,m-1,\ldots,0}\) up to cyclic reordering (a result of Milnor) and the choice of base points \(x_j\) is proved by using the technics developped in the paper.