  In this article, we study wave dynamics in the fractional nonlinear
Schr\"odinger equation with a modulated honeycomb potential. This problem
arises from recent research interests in the interplay between topological
materials and nonlocal governing equations. Both are current focuses in
scientific research fields. We first develop the Floquet-Bloch spectral theory
of the linear fractional Schr\"odinger operator with a honeycomb potential.
Especially, we prove the existence of conical degenerate points, i.e., Dirac
points, at which two dispersion band functions intersect. We then investigate
the dynamics of wave packets spectrally localized at a Dirac point and derive
the leading effective envelope equation. It turns out the envelope can be
described by a nonlinear Dirac equation with a varying mass. With rigorous
error estimates, we demonstrate that the asymptotic solution based on the
effective envelope equation approximates the true solution well in the
weighted-$H^s$ space.
