  In the presence of a static quark--antiquark pair, the spectrum of the
low-lying states in SU($N$) gauge theories is discrete and likely to be
described, at large quark separations $r$, by an effective string theory. The
expansion of the excitation energies in powers of $1/r$, which derives from the
latter, involves an increasing number of unknown couplings that characterize
the string self-interactions. Using open--closed string duality, we show that
the possible values of the couplings are constrained by a set of algebraic
relations. In particular, the corrections of order $1/r^2$ must vanish, while
the $1/r^3$ terms (which we work out for the few lowest levels) depend on a
single adjustable coupling only.
