Perturbed β-γ systems and complex geometry

We consider the equations, arising as the conformal invariance conditions of the perturbed curved betagamma 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. (C) 2007 Elsevier B.V. All rights reserved.