Exciton band structure of monolayer MoS$_2$

We describe a theory of the momentum-dependent exciton spectrum of monolayer molybdenum disulfide. Low-energy excitons occur both at the Brillouin zone center and at the Brillouin-zone corners. We find that binding energies at the Brillouin-zone center deviate qualitatively from the $(n-1/2)^{-2}$ pattern of the two-dimensional hydrogenic model. Moreover, the four $2p$ states of $A$ series are lower in energy than the corresponding $2s$ states and not degenerate. The two-fold ground-state valley degeneracy is lifted linearly at small momenta by electron-hole exchange processes that establish inter valley coherence. We conclude that atlhough monolayer MoS$_2$ is a direct-gap semiconductor when classified by its quasiparticle band structure it may well be an indirect gap material when classified by its excitation spectra, and speculate on the role of this property in luminescence characteristics.