## Homology, Homotopy and Applications

### Holohonies for connections with values in $L_\infty$-algebras

#### Abstract

Given a flat connection $\alpha$ on a manifold $M$ with values in a filtered $L_\infty$-algebra $\mathfrak{g}$, we construct a morphism $\mathsf{hol}^{\infty}_\alpha \colon C_\bullet(M) \rightarrow \mathsf{B} \hat{\mathbb{U}}_\infty(\mathfrak{g})$, which generalizes the holonomy map associated to a flat connection with values in a Lie algebra. The construction is based on Gugenheim's $\mathsf{A}_{\infty}$-version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space $\mathbb{R}^d$, and to generalizations of the holonomy representations of braid groups.

#### Article information

Source
Homology Homotopy Appl., Volume 16, Number 1 (2014), 89-118.

Dates
First available in Project Euclid: 3 June 2014

Abad, Camilo Arias; Schätz, Florian. Holohonies for connections with values in $L_\infty$-algebras. Homology Homotopy Appl. 16 (2014), no. 1, 89--118. https://projecteuclid.org/euclid.hha/1401800074