Perturbed Beta-Gamma Systems and Complex Geometry

TL;DR

This paper explores the equations from perturbed beta-gamma systems, linking them to Einstein equations with additional fields on hermitian manifolds, revealing their geometric and algebraic structures.

Contribution

It demonstrates the decomposition of these equations into linear and bilinear parts, connecting them to generalized Maurer-Cartan structures and Courant brackets.

Findings

Equations imply the vanishing of the first Chern class.

Relation established between equations and generalized Maurer-Cartan structures.

Connections made between bilinear operations and Courant/Dorfman brackets.

Abstract

We consider the equations, arising as the conformal invariance conditions of the perturbed curved beta-gamma system. These equations have the physical meaning of Einstein equations with a B-field and a dilaton on a hermitian manifold, where the B-field 2-form is imaginary and proportional to the canonical form associated with hermitian metric. We show that they decompose into linear and bilinear equations and lead to the vanishing of the first Chern class of the manifold where the system is defined. We discuss the relation of these equations to the generalized Maurer-Cartan structures related to BRST operator. Finally we describe the relations of the generalized Maurer-Cartan bilinear operation and the Courant/Dorfman brackets.