  We investigate the lowest many-body excited states in carbon nanotubes by
means of ab initio calculations. On the basis of these calculations and an
additional theoretical analysis of the excitons, we demonstrate that the
splitting between the dark and bright spin-singlet states is due to the direct
Coulomb electron-hole interaction, that the dark exciton always has gerade
symmetry, and that it is always lower in energy relative to the bright exciton.
Applying these studies to 0.4-0.8-nm-diameter tubes, we find a gerade-ungerade
splitting of several tens of an meV, and a singlet-triplet splitting of
comparable value. Our symmetry analysis of the origin of the singlet splitting
indicates possible startegies to increase the emission efficiency of nanotubes.
