  We present the first experimental constraints on the spin-dependent
WIMP-nucleon elastic cross sections from LUX data acquired in 2013. LUX is a
dual-phase xenon time projection chamber operating at the Sanford Underground
Research Facility (Lead, South Dakota), which is designed to observe the recoil
signature of galactic WIMPs scattering from xenon nuclei. A profile likelihood
ratio analysis of $1.4~\times~10^{4}~\text{kg}\cdot~\text{days}$ of fiducial
exposure allows 90% CL upper limits to be set on the WIMP-neutron (WIMP-proton)
cross section of $\sigma_n~=~9.4~\times~10^{-41}~\text{cm}^2$
($\sigma_p~=~2.9~\times~10^{-39}~\text{cm}^2$) at 33 GeV/c$^2$. The
spin-dependent WIMP-neutron limit is the most sensitive constraint to date.
