  We calculate linear and nonlinear optical susceptibilities arising from the
excitonic states of mono- layer MoS2 for in-plane light polarizations, using
second-quantized bound and unbound exciton operators. Optical selection rules
are critical for obtaining the susceptibilities. We derive the valley-chirality
rule for the second harmonic generation in monolayer MoS2, and find that the
third- harmonic process is efficient only for linearly polarized input light
while the third-order two photon process (optical Kerr effect) is efficient for
circularly polarized light using a higher order exciton state. The absence of
linear absorption due to the band gap and the unusually strong two-photon
third-order nonlinearity make the monolayer MoS2 excitonic structure a
promising resource for coherent nonlinear photonics.
