BRST, Generalized Maurer-Cartan Equations and CFT

TL;DR

This paper explores the algebraic structure of BRST charge conservation in perturbed conformal field theories, revealing that these conditions can be expressed as generalized Maurer-Cartan equations, with applications to sigma models and Einstein equations.

Contribution

It formulates BRST conservation laws in perturbed CFTs as generalized Maurer-Cartan equations using operator algebra and OPE, extending the understanding of conformal invariance conditions.

Findings

Derived algebraic equations for BRST charge conservation

Interpreted Einstein equations as generalized Maurer-Cartan equations

Analyzed examples: bosonic sigma model and first order theory

Abstract

The paper is devoted to the study of BRST charge in perturbed two dimensional conformal field theory. The main goal is to write the operator equation expressing the conservation law of BRST charge in perturbed theory in terms of purely algebraic operations on the corresponding operator algebra, which are defined via the OPE. The corresponding equations are constructed and their symmetries are studied up to the second order in formal coupling constant. It appears that the obtained equations can be interpreted as generalized Maurer-Cartan ones. We study two concrete examples in detail: the bosonic nonlinear sigma model and perturbed first order theory. In particular, we show that the Einstein equations, which are the conformal invariance conditions for both these perturbed theories, expanded up to the second order, can be rewritten in such generalized Maurer-Cartan form.