  In the local universe, galaxies in clusters show different properties
compared to more isolated systems. Understanding how this difference originates
and whether it is already in place at high redshift is still a matter of
debate. Thanks to uniquely deep optical spectra from the VANDELS survey, we
investigate environmental effects on the stellar mass-metallicity relation
(MZR) for a sample of ~1000 star-forming galaxies at 2<z<4. We complement our
dataset with MOSFIRE follow-up of 21 galaxies to study the environmental
dependence of the gas-phase MZR. Robust stellar and gas metallicities are
derived, respectively, from well-calibrated photospheric absorptions features
at 1501 and 1719 \AA in the stacked spectra, and from optical emission lines
([OII]3726-3729, [OIII]5007, and Hbeta) in individual systems. We characterize
the environment through multiple criteria by using the local galaxy density
maps previously derived in VANDELS. We find that environmental effects are weak
at these redshifts, and more important around the densest overdensity
structures, where galaxies have a lower stellar Z (by 0.2 dex) and a lower
gas-phase Z (by 0.1 dex) compared to the field, with a significance of 1 and 2
sigma, respectively. Crucially, this offset cannot be explained by a selection
effect due to a higher SFR, a fainter UV continuum, or different dust
attenuations and stellar ages. Despite the still low S/N of our results, we
propose a combination of increased mergers and high-speed encounters, more
efficient AGN feedback in dense cores, and cold gas inflows as viable
mechanisms diluting the metal content of overdense galaxies or expelling their
metals to the IGM. Finally, some tensions remain with semi-analytic models and
hydrodynamical simulations, which predict no significant offset as a function
of host halo mass, suggesting that an explicit implementation of environmental
processes is needed.
