Selfduality of [formula omitted] reduction of gravity coupled to a σ-model

Abstract

Dimensional reduction in two dimensions of gravity in higher dimension, or more generally of d=3 gravity coupled to a σ-model on a symmetric space, is known to possess an infinite number of symmetries. We show that such a bidimensional model can be embedded in a covariant way into a σ-model on an infinite symmetric space, built on the semidirect product of an affine group by the Witt group. The finite theory is the solution of a covariant selfduality constraint on the infinite model. It has therefore the symmetries of the infinite symmetric space. (We give explicit transformations of the gauge algebra.) The usual physical fields are recovered in a triangular gauge, in which the equations take the form of the usual linear systems which exhibit the integrable structure of the models. Moreover, we derive the constraint equation for the conformal factor, which is associated to the central term of the affine group involved.