Maurer-Cartan methods in deformation theory: the twisting procedure

TL;DR

This monograph explores Maurer-Cartan methods in deformation theory, focusing on the twisting procedure that constructs new algebraic structures from Maurer-Cartan elements, with applications across mathematics and physics.

Contribution

It introduces a simplified presentation of the twisting procedure for operads, broadening the scope of examples and applications in graph homology and deformation theory.

Findings

Provides a criterion for quadratic operads to admit twisting procedures

Introduces a new, simpler approach to operad twisting à la Willwacher

Recovers and extends known graph complexes in deformation theory

Abstract

This monograph provides an overview on the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a conceptual, exhaustive and gentle treatment of the twisting procedure, which functorially creates new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer-Cartan element. The twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras is described by means of the action of the biggest deformation gauge group ever considered. We give a criterion on quadratic operads for the existence of a meaningful twisting procedure of their associated categories of algebras. And, we introduce the twisting procedure for operads \`a la Willwacher using a new and simpler presentation, which provides us with a wide source of motivating examples related to graph homology, both recovering known graph complexes (due to Kontsevich) and introducing some new ones. This book starts with elementary surveys on gauge theory and deformation theory using differential graded Lie algebras in order to ease the way to the theory. It finishes with concise surveys on the fundamental theorem of deformation theory, higher Lie theory, rational homotopy theory, simplicial theory of homotopy algebras, and the Floer cohomology of Lagrangian submanifolds, to illustrate deep examples of applications.