  We show that band topology can dramatically change the photophysics of
two-dimensional (2D) semiconductors. For systems in which states near the band
extrema are of multiple orbitals character and the spinors describing the
orbital components (pseudospins) pick up nonzero winding numbers (topological
invariants) around the extremal k-point, the optical strength and nature (i.e.,
helicity) of the excitonic states are dictated by the optical matrix element
winding number, a unique and heretofore unrecognized characteristic. We
illustrate these findings in three gapped graphene systems - monolayer graphene
with inequivalent sublattices and biased bi- and tri-layer graphene, where the
pseudospin textures manifest into a unique optical matrix element winding
pattern associated with different valley and photon circular polarization. This
winding-number physics leads to novel exciton series and optical selection
rules, with each valley hosting multiple bright excitons coupled to light of
different helicity. This valley-exciton selective circular dichroism can be
unambiguously detected using optical spectroscopy.
