Unified Einstein–Virasoro Master Equation in the General Nonlinear Sigma Model

Abstract

The Virasoro master equation (VME) describes the general affine-Virasoro construction T = LabJa Jb + iDa ∂ Ja in the operator algebra of the WZW model, where Lab is the inverse inertia tensor and Da is the improvement vector. In this paper, we generalize this construction to find the general (one-loop) Virasoro construction in the operator algebra of the general nonlinear sigma model. The result is a unified Einstein–Virasoro master equation which couples the space–time spin-2 field Lab to the background fields of the sigma model. For a particular solution [Formula: see text], the unified system reduces to the canonical stress tensors and conventional Einstein equations of the sigma model, and the system reduces to the general affine-Virasoro construction and the VME when the sigma model is taken to be the WZW action. More generally, the unified system describes a space of conformal field theories which is presumably much larger than the sum of the general affine-Virasoro construction and the sigma model with its canonical stress tensors. We also discuss a number of algebraic and geometrical properties of the system, including its relation to an unsolved problem in the theory of G-structures on manifolds with torsion.