Unified Einstein-Virasoro Master Equation in the General Non-Linear Sigma Model

The Virasoro master equation (VME) describes the general affine-Virasoro construction T = LJaJb + iD ∂Ja in the operator algebra of the WZW model, where L ab is the inverse inertia tensor and D 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 spacetime spin-two field L to the background fields of the sigma model. For a particular solutionL G , 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. CERN-TH/96-132 April 1996 ∗e-mail address: [email protected] †On leave from the Department of Physics, University of California, Berkeley, CA 94720, USA. ‡e-mail address: [email protected], [email protected]