  Cosmic shear is a powerful probe of cosmology, but it is affected by the
intrinsic alignment (IA) of galaxy shapes with the large-scale structure.
Upcoming surveys like Euclid and Vera C. Rubin Observatory's Legacy Survey of
Space and Time (LSST) require an accurate understanding of IA, particularly for
higher-order cosmic shear statistics that are vital for extracting the most
cosmological information. In this paper, we report the first detection of
third-order IA correlations using the LOWZ galaxy sample from the Sloan Digital
Sky Survey (SDSS) Baryon Oscillation Spectroscopic Survey (BOSS). We compare
our measurements with predictions from the MICE cosmological simulation and an
analytical NLA-inspired model informed by second-order correlations. We also
explore the dependence of the third-order correlation on the galaxies'
luminosity. We find that the amplitude $A_\mathrm{IA}$ of the IA signal is
non-zero at the $4.7\sigma$ ($7.6\sigma$) level for scales between $6 h^{-1}
\mathrm{Mpc}$ ($1 h^{-1} \mathrm{Mpc}$) and $20 h^{-1} \mathrm{Mpc}$. For
scales above $6 h^{-1}\mathrm{Mpc}$ the inferred AIA agrees both with the
prediction from the simulation and estimates from second-order statistics
within $1\sigma$ but deviations arise at smaller scales. Our results
demonstrate the feasibility of measuring third-order IA correlations and using
them for constraining IA models. The agreement between second- and third-order
IA constraints also opens the opportunity for a consistent joint analysis and
IA self-calibration, promising tighter parameter constraints for upcoming
cosmological surveys.
