Formal Maurer-Cartan Structures: from CFT to Classical Field Equations

TL;DR

This paper reformulates classical field equations like Einstein and Yang-Mills as generalized Maurer-Cartan equations using BRST operators, revealing their symmetries and underlying algebraic structures.

Contribution

It introduces a novel formulation of classical field equations as GMC equations with BRST differential, connecting gauge and diffeomorphism symmetries to algebraic structures.

Findings

Classical field equations are expressed as GMC equations.

Symmetries correspond to diffeomorphism and gauge invariance.

Properties of bilinear operations in GMC are studied.

Abstract

We show how the well-known classical field equations as Einstein and Yang-Mills ones, which arise as the conformal invariance conditions of certain two-dimensional theories, expanded up to the second order in the formal parameter, can be reformulated as Generalized/formal Maurer-Cartan equations (GMC), where the differential is the BRST operator of String theory. We introduce the bilinear operations which are present in GMC, and study their properties, allowing us to find the symmetries of the resulting equations which will be naturally identified with the diffeomorphism and gauge symmetries of Einstein and Yang-Mills equations correspondingly.