We describe recent results of optical experiments on single-walled carbon nanotubes in high magnetic fields, probing the influence of a tube-threading magnetic flux on their band structure and excitonic states. The magnetic flux breaks the time-reversal symmetry and thus lifts the K-K′ valley degeneracy, and the amount of state splitting is determined by the Aharonov-Bohm phase. We show experimental evidence that this field-induced symmetry breaking overcomes the Coulomb-induced exciton splitting which is predicted to make the lowest singlet exciton state optically inactive (or "dark"). Thus, a magnetic field applied parallel to the tube axis "brightens" the dark exciton, resulting in a drastic increase in photoluminescence intensity with magnetic field. We also find that the amount of brightening increases with decreasing temperature.