Construction of Maurer-Cartan elements over configuration spaces of curves

Benjamin Enriquez (1) , Federico Zerbini (2, 3)

1 IRMA - Institut de Recherche Mathématique Avancée
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681
3 Université Paris-Saclay

For C a complex curve and n ≥ 1, a pair ($P$, $\Delta_P$) of a principal bundle $P$ with meromorphic flat connection over $C^n$ , holomorphic over the configuration space $C_n(C)$ of n points over $C$, was introduced in [En]. For any point ∞ ∈ C, we construct a trivialisation of the restriction of P to (C \ ∞)$^n$ and obtain a Maurer-Cartan element $J$ over $C_n$(C \ ∞) out of $\Delta_P$ , 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 $\Delta_P$. When n = 1, this construction gives rise to elements of Hain's space of second kind iterated integrals over C.

Domaines: Géométrie algébrique [math.AG]

Benjamin Enriquez, Federico Zerbini. Construction of Maurer-Cartan elements over configuration spaces of curves. 2021. ⟨cea-03406087⟩