Closed smooth strings on a torus

The free energy of a gas of closed strings with extrinsic curvature (smooth strings) is evaluated on a torus. This is compared with the free energy of a collection of free particles, and hence the mass spectrum of excitations of the smooth strings is deduced. It is found that above a critical value of the coupling constant of the curvature term the spectrum is free of tachyons. Furthermore, there are no massless spin-2 excitations. The absence of massless spin-2 fields is a consequence of the fact that the smooth-string theory is not modular invariant.