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Construction of Maurer-Cartan elements over configuration spaces of curves
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abstract
For $C$ a complex curve and $n \geq 1$, 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 $C$.
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