On a Zero Curvature Representation for Bosonic Strings and $p$-Branes

Abstract

It is shown that a zero curvature representation for $D$-- dimensional $p$-- brane equations of motion originates naturally in the geometric (Lund- Regge- Omnes) approach. To study the possibility to use this zero curvature representation for investigation of nonlinear equations of $p$-- branes, the simplest case of $D$-- dimensional string ($p=1$) is considered. The connection is found between the $SO(1,1)$ gauge (world--sheet Lorentz) invariance of the string theory with a nontrivial dependence on a spectral parameter of the Lax matrices associated with the nonlinear equations describing the embedding of a string world sheet into flat $D$-- dimensional space -- time. Namely, the spectral parameter can be identified with a parameter of constant $SO(1,1)$ gauge transformations, after the deformation of the Lax matrices has been performed.