E., Rigby, J., et al

We measure the gas-phase abundances of the elements He, N, O, Ne, S, Ar, and Fe in an individual H\textsc{ii} region known to be leaking Lyman-continuum photons in the Sunburst Arc, a highly magnified galaxy at redshift $z=2.37$. We detect the temperature-sensitive auroral lines [SII]$\lambda\lambda4069,4076$, [OII]$\lambda\lambda7320,7330$, [SIII]$\lambda6312$, [OIII]$\lambda4363$, and [NeIII]$\lambda3343$ in a stacked spectrum of 5 multiple images of the Lyman-continuum emitter (LCE), from which we directly measure the electron temperature in the low, intermediate, and high ionization zones. We also detect the density-sensitive doublets of [OII]$\lambda\lambda3727,3729$, [SII]$\lambda\lambda6717,6731$, and [ArIV]$\lambda\lambda4713,4741$, which constrain the density in both the low- and high-ionization gas. With these temperature and density measurements, we measure gas-phase abundances with similar rigor as studies of local galaxies. We measure a gas-phase metallicity for the LCE of $12+\log(\textrm{O}/\textrm{H}) = 7.97 \pm 0.05$, and find an enhanced nitrogen abundance $\log(\textrm{N}/\textrm{O}) = -0.65^{+0.16}_{-0.25}$. This nitrogen abundance is consistent with enrichment from a population of Wolf-Rayet stars, additional signatures of which are reported in a companion paper. Abundances of sulfur, argon, neon, and iron are consistent with local low-metallicity H\textsc{ii} regions and low-redshift galaxies. This study represents the most complete chemical abundance analysis of an individual H\textsc{ii} region at Cosmic Noon to date, which enables direct comparisons between local H\textsc{ii} regions and those in the distant universe.