Construction of Maurer-Cartan elements over configuration spaces of curves

Abstract

For $C$ a complex curve and $n\ge 1$, a pair $(\mathcal{P},{\nabla }_{\mathcal{P}})$ of a principal bundle $\mathcal{P}$ with meromorphic flat connection over $C^n$, holomorphic over the configuration space $C_n(C)$ of $n$ points over $C$, was introduced in arXiv:1112.0864. For any point $\infty \in C$, we construct a trivialisation of the restriction of $\mathcal{P}$ to $(C\setminus \infty {)}^n$ and obtain a Maurer-Cartan element $J$ over $C_n(C\setminus \infty )$ out of ${\nabla }_{\mathcal{P}}$, thus generalising a construction of Levin and Racinet when the genus of $C$ is higher than one. We give explicit formulas for $J$ as well as for ${\nabla }_{\mathcal{P}}$. When $n=1$, this construction gives rise to elements of Hain's space of second kind iterated integrals over $C$.

50 pages, 4 figures

References in corpus (1)

• Towards multiple elliptic polylogarithms