Homology, Homotopy and Applications

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

Camilo Arias Abad and Florian Schätz

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 3 June 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18G55: Homotopical algebra 55R65: Generalizations of fiber spaces and bundles

Iterated integral higher holonomy $L_\infty$-algebra configuration space braid group


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

Export citation