1 Introduction

In the large-$N$ limit $SU(N)$ Yang-Mills gauge theories display string-like excitations [1]. The effective dynamics of these one-dimensional objects is described by a Schild action which is invariant under area preserving reparametrizations only. This result allowed to establish a relationship between $SU(\, \infty\,)$ and symplectic transformations: $\sigma^m\to \sigma^{\prime\,m}= \sigma^{\prime\, m}(\sigma)$, $\vert\, {\partial\sigma^\prime \over \partial\sigma}\,\vert=1$, but not between $SU(\, \infty\,)$ and the group of general reparametrization encoded into the Nambu-Goto action.
In this letter we are going to show how to recover the reparametrization invariant Nambu-Goto action form the large-$N$ limit of a $SU(N)$ gauge invariant action.