Tachyons and misaligned supersymmetry in closed string vacua

In a remarkable paper, Dienes discovered that the absence of physical tachyons in closed string theory is intimately related to oscillations in the net number of bosonic minus fermionic degrees of freedom, a pattern predicted by an underlying misaligned supersymmetry. The average of these oscillations was linked to an exponential growth controlled by an effective central charge C-eff smaller than the expected inverse Hagedorn temperature. Dienes also conjectured that C-eff should vanish when tachyons are absent. In this paper, we revisit this problem and show that boson-fermion oscillations are realised even when tachyons are present in the physical spectrum. In fact, we prove that the average growth rate C-eff is set by the mass of the "lightest" state, be it massless or tachyonic, and coincides with the effective inverse Hagedorn temperature of the associated thermal theory. We also provide a general proof that the necessary and sufficient condition for classical stability is the vanishing of the sector averaged sum which implies C-eff = 0, in agreement with Dienes' conjecture.