Pre-Lie deformation theory

TL;DR

This paper develops a deformation theory for pre-Lie algebras, introducing a new integration method, and applies it to homotopy algebra structures, providing a conceptual proof of the Homotopy Transfer Theorem.

Contribution

It introduces a novel integration theory for pre-Lie algebras and offers a homotopical description of the Deligne groupoid, advancing the understanding of homotopy algebra structures.

Findings

Provides a new integration theory for pre-Lie algebras

Offers a homotopical description of the Deligne groupoid

Proves the Homotopy Transfer Theorem using gauge action

Abstract

In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the $dd^c$-lemma.