Membranes and higher groupoids

TL;DR

This paper explores the structure of dg-Lie algebras generated by universal flat dg-connections on affine spaces, generalizing classical theorems and linking to higher-dimensional holonomy and membranes.

Contribution

It introduces the semiabelianization of dg-Lie algebras related to universal connections and demonstrates their role in higher-dimensional holonomy and membrane theory.

Findings

Description of semiabelianization in terms of closed differential forms

Generalization of Reutenauer's abelianization theorem

Establishment of higher-dimensional holonomy via dg-Lie algebra connections

Abstract

We study the dg-Lie algebra f_n generated by the coefficients of the universal translation invariant flat dg-connection on the n-dimensional affine space. We describe its "semiabelianization" (in particular, the universal quotient which is a crossed module of Lie algebras) in terms of closed differential forms of arbitrary order. This generalizes the theorem of Reutenauer on the abelianization of the commutant of a free Lie algebra. Semiabelian dg-Lie algebras, i.e., non-poisitively graded dg-Lie algebras with brackets of any elements of strictly negative degree being 0, are Lie algebraic analogs of crossed complexes of Brown-Higgins. We show that dg-connections with values in such dg-Lie algebras give rise to higher-dimensional holonomy in the strict sense, i.e., to a system of data associating an object of a strict higher category to a geometric membrane (a class of parametrized membranes up to thin homotopy). Applied to canonical connection with coefficients in the semiabelianization of f_n, this gives a way to assiociate a power-series-type object to a geometric membrane in R^n.