MARTA: Temperature-temperature relationships and strong-line metallicity calibrations from multiple auroral-line detections at cosmic noon

We present the first results from MARTA (Measuring Abundances at high Redshift with the T${}_e$ Approach), a programme leveraging ultra-deep, medium-resolution JWST/NIRSpec spectroscopy to probe the interstellar medium (ISM) of star-forming galaxies at $z\sim 2-3$. We report detections of one or more auroral lines, including [O III]$\lambda 4363$, [O II]$\lambda \lambda 7320,7330$, [S II] $\lambda 4068$, and [S III] $\lambda 6312$ for 16 galaxies in the sample, providing measurements of multiple ionic temperatures. We tested the validity of the T[O II]-T[O III] relation at high redshift considering a total sample of 21 objects including literature data, and obtained a shallower slope than in the low-$z$ literature. However, such a slope is consistent with low-redshift data when ultra-low metallicity objects are considered. We assessed the correlation of the T[O II]-T[O III] relationship and its scatter on different physical parameters, finding a mild correlation with the ionisation parameter and radiation field hardness, while no significant correlation with gas density. The location of high-redshift data is also consistent with the low-$z$ literature in the T[O II]-T[S II], and T[S III]-T[O III] relations, although this conclusion is limited with low-number statistics. Finally, we leveraged our sample together with a comprehensive compilation of galaxies with [O III]$\lambda 4363$ detections from the literature to recalibrate classical strong-line diagnostics at high redshift. MARTA represents a key addition in this space because it provides direct metallicities at moderately high oxygen abundances (12+log(O/H) $\sim 8.0-8.4$).