Maurer-Cartan elements and homotopical perturbation theory

TL;DR

This paper explores the relationship between Maurer-Cartan elements in curved L-infinity algebras and homotopical perturbation theory, establishing a bijection under certain conditions.

Contribution

It proves a bijection between Maurer-Cartan elements in a curved L-infinity algebra and those in a subcomplex, extending Fukaya's construction.

Findings

Bijection between Maurer-Cartan elements in L and M under projection

Extension of Fukaya's construction to curved L-infinity algebras

New insights into homotopical perturbation theory applications

Abstract

Let L be a (pro-nilpotent) curved L-infinity algebra, and let h be a homotopy between L and a subcomplex M. Using homotopical perturbation theory, Fukaya constructed from this data a curved L-infinity structure on M. We prove that projection from L to M induces a bijection between the set of Maurer-Cartan elements x of L such that hx=0 and the set of Maurer-Cartan elements of M.