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

TL;DR

This paper demonstrates that a zero curvature representation naturally arises in the geometric approach to $p$-brane equations, specifically analyzing the case of strings and linking gauge invariance to spectral parameters in Lax matrices.

Contribution

It establishes a connection between gauge invariance and spectral parameters in the zero curvature representation for string equations of motion.

Findings

Zero curvature representation is naturally linked to the geometric approach.

Spectral parameter relates to $SO(1,1)$ gauge transformations.

Connection between gauge invariance and Lax matrix deformation.

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.