  During the last decades, rapid progress has been made in measurements of the
rest-frame ultraviolet (UV) luminosity function (LF) for high-redshift galaxies
($z \geq 6$). The faint-end of the galaxy LF at these redshifts provides
powerful constraints on different dark matter models that suppress small-scale
structure formation. In this work we perform full hydrodynamical cosmological
simulations of galaxy formation using an alternative DM model composed of
extremely light bosonic particles ($m \sim 10^{-22}$ eV), also known as fuzzy
dark matter (FDM), and examine the predictions for the galaxy stellar mass
function and luminosity function at $z \geq 6$ for a range of FDM masses. We
find that for FDM models with bosonic mass $m = 5\times10^{-22}$ eV, the number
density of galaxies with stellar mass $\rm M_* \sim 10^7 M_{\odot}$ is
suppressed by $\sim 40\%$ at z = 9, $\sim 20\%$ at z = 5, and the UV LFs within
magnitude range of -16 < $M_{\rm UV}$ < -14 is suppressed by $\sim 60\%$ at $z
= 9$, $\sim 20\%$ at $z = 5$ comparing to the CDM counterpart simulation.
Comparing our predictions with current measurements of the faint-end LFs ($-18
\leqslant M_{\rm UV} \leqslant -14$), we find that FDM models with $m_{22} <
5\times10^{-22}$ are ruled out at $3\sigma$ confidence level. We expect that
future LF measurements by James Webb Space Telescope (JWST), which will extend
down to $M_{\rm UV} \sim -13$ for $z \lesssim 10$, with a survey volume that is
comparable to the Hubble Ultra Deep Field (HUDF) would have the capability to
constrain FDM models to $m\; \gtrsim 10^{-21}$ eV.
