  We report high field magneto-optical study on the first and second sub-band
transitions of single-chirality single-walled carbon nanotubes. The ordering
and relative energy splitting between bright and dark excitonic states were
found to be inverse between the first and second subbands. We verified that the
zero-momentum dark singlet exciton lies below the bright exciton for the first
subband transitions, while for the second sub-band transitions, it was found to
have higher energy than the bright excitonic state. Effect of this peculiar
excitonic structure was found to manifest itself in distinctive Aharonov-Bohm
splitting in ultra-high magnetic fields up to 190 T.
