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String excitation energies in SU(N) gauge theories beyond the free-string approximation

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.