Lie's Third Theorem for Lie $\infty$-Algebras

TL;DR

This paper proves Lie's third theorem for finite-type Lie infinity-algebras, showing they integrate into Lie infinity-groups, and introduces local minimal models for Kan simplicial manifolds to facilitate this integration.

Contribution

It provides the first proof of Lie's third theorem for finite-type Lie infinity-algebras and introduces local minimal models for Kan simplicial manifolds.

Findings

Finite-type Lie $$-algebras integrate into Lie $$-groups.

Constructs a new finite-dimensional model for the string Lie 2-group.

Establishes local minimal models for Kan simplicial manifolds.

Abstract

We introduce the theory of local minimal models for Kan simplicial manifolds, which provide the appropriate generalization of minimal Kan simplicial sets to geometric contexts. We use this to obtain the first proof of Lie's third theorem for finite-type Lie $\infty$-algebras: Every finite-type, homologically and non-negatively graded $L_\infty$-algebra over $\mathbb{R}$ integrates to a finite-dimensional Lie $\infty$-group. As a corollary, our construction yields a new explicit finite-dimensional model for the string Lie 2-group.