Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction of Maurer-Cartan elements over configuration spaces of curves

Benjamin Enriquez, Federico Zerbini

TL;DR

This work constructs Maurer–Cartan elements on configuration spaces of curves by marrying a holomorphic flat connection with a universal Schottky-uniformized framework. The authors develop a detailed trivialization of a principal bundle over (C\setminus∞)^n, introducing auxiliary functions and a pro-nilpotent Lie algebra to encode monodromy data via iterated integrals. They define a canonical 1-form K and gauge-transformations g that transport the flat connection to a Maurer–Cartan form J on C_n(C\setminus∞), and they analyze the dependence of J on auxiliary data, providing decompositions to separate multi-valued and single-valued components. The paper culminates with algorithms to compute these objects explicitly, including low-degree formulas, enabling concrete calculations of iterated integrals and second-kind-type data in higher-genus settings. This advances understanding of pro-unipotent fundamental groups of configuration spaces and connects to Hain’s iterated integrals and Levin–Racinet-type constructions in a genus-general setting.

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 $C$.

Construction of Maurer-Cartan elements over configuration spaces of curves

TL;DR

This work constructs Maurer–Cartan elements on configuration spaces of curves by marrying a holomorphic flat connection with a universal Schottky-uniformized framework. The authors develop a detailed trivialization of a principal bundle over (C\setminus∞)^n, introducing auxiliary functions and a pro-nilpotent Lie algebra to encode monodromy data via iterated integrals. They define a canonical 1-form K and gauge-transformations g that transport the flat connection to a Maurer–Cartan form J on C_n(C\setminus∞), and they analyze the dependence of J on auxiliary data, providing decompositions to separate multi-valued and single-valued components. The paper culminates with algorithms to compute these objects explicitly, including low-degree formulas, enabling concrete calculations of iterated integrals and second-kind-type data in higher-genus settings. This advances understanding of pro-unipotent fundamental groups of configuration spaces and connects to Hain’s iterated integrals and Levin–Racinet-type constructions in a genus-general setting.

Abstract

For a complex curve and , a pair of a principal bundle with meromorphic flat connection over , holomorphic over the configuration space of points over , was introduced in arXiv:1112.0864. For any point , we construct a trivialisation of the restriction of to and obtain a Maurer-Cartan element over out of , thus generalising a construction of Levin and Racinet when the genus of is higher than one. We give explicit formulas for as well as for . When , this construction gives rise to elements of Hain's space of second kind iterated integrals over .

Paper Structure

This paper contains 53 sections, 58 theorems, 141 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Contents of the article
  4. Trivialization of the restriction of $\mathcal{P}$ to $(C\setminus\infty)^n$
  5. The geometric setup
  6. Fundamental group and universal cover
  7. Fundamental groups of Riemann surfaces
  8. 1-forms on $C$ and iterated integrals
  9. 1-forms on $C$
  10. Iterated integrals
  11. Groupoids
  12. 1-nerves
  13. Quotients
  14. The groupoid $\Pi_1(X)$
  15. The groupoid morphism $F_{\pmb\Lambda} : \Pi_1(C\setminus\infty)\to \mathrm{exp}(\hat{\mathfrak G})$
  16. ...and 38 more sections

Key Result

Lemma 2.2

(a) There is a right action of $\pi_1(X,x)$ on $p^{-1}(x)$, denoted $(\tilde{x},\eta)\mapsto \tilde{x}\cdot\eta$, where $\tilde{x}\cdot\eta$ is the endpoint of the unique lift starting at $\tilde{x}$ of any representative of $\eta$. (b) There is a unique map $p^{-1}(x)\times\pi_1(X,x)\to\mathrm{Aut}

Figures (4)

  • Figure 1: The domain $D\subset C$ as the interior of a polygon with boundary identifications.
  • Figure 2: The domain $\tilde{D}\subset \tilde{C}$.
  • Figure 3: The domain $\hat{D}\subset \hat{C}$ as the interior of a polygon with boundary identifications.
  • Figure 4: The external boundary should be thought of as the point $\hat{\infty}$. The domain $\hat{D}\subset \hat{C}$ is the complement of the coloured regions.

Theorems & Definitions (131)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 121 more