Self-intersection of curves in surfaces and Drinfeld associators

Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb{Q}[\pi]$ of the fundamental group $\pi$ of a surface with boundary [1]. The first operation is binary and measures the intersection of two oriented curves on the surface, while the second operation is unary and computes the self-intersection of an oriented curve. It is already known that Turaev's intersection pairing has a simple algebraic description when the $I$-adic completion of the group algebra $\mathbb{Q}[\pi]$ is appropriately identified to the degree-completion of the tensor algebra $T(H)$ of $H:=H_1(\pi;\mathbb{Q})$.
We will show that Turaev's self-intersection map has a similar description in the case of a disk with $p$ punctures. In this special case, we will consider those identifications between the completions of $\mathbb{Q}[\pi]$ and $T(H)$ that arise from the Kontsevich integral by embedding $\pi$ into the pure braid group on $(p+1)$ strands [2, 3]. As a matter of fact, our algebraic description involves a formal power series which is explicitly determined by the Drinfeld associator $\Phi$ entering into the definition of the Kontsevich integral; this series is essentially Enriquez' $\Gamma$-function of $\Phi$ [4]. If time allows, we will also discuss the case of higher-genus surfaces. (This talk is based on the preprint [5].)
References:

• V. Turaev, Intersections of loops in two-dimensional manifolds. (Russian) Mat. Sb. 106(148) (1978), no. 4, 566–588. English translation: Math. USSR–Sb. 35 (1979), 229–250.
• N. Habegger, G. Masbaum, The Kontsevich integral and Milnor's invariants. Topology 39 (2000), no. 6, 1253–1289.
• A. Alekseev, B. Enriquez, C. Torossian, Drinfeld associators, braid groups and explicit solutions of the Kashiwara–Vergne equations. Publ. Math. Inst. Hautes Études Sci. 112 (2010), 143–189.
• B. Enriquez, On the Drinfeld generators of $\mathfrak{grt}_1(\mathbf{k})$ and $\Gamma$-functions for associators. Math. Res. Lett. 3 (2006), no. 2-3, 231–243.
• G. Massuyeau, Formal descriptions of Turaev's loop operations. Preprint (2015), arXiv:1511.03974.