The Rest-Frame Optical Spectroscopic Properties of Ly$\alpha$-Emitters
  at $z\sim2.5$: The Physical Origins of Strong Ly$\alpha$ Emission
  We present rest-frame optical spectra of 60 faint ($R_{AB}\sim 27$; $L\sim0.1
L_*$) Ly$\alpha$-selected galaxies (LAEs) at $z\approx2.56$. The average LAE is
consistent with the extreme low-metallicity end of the continuum-selected
galaxy distribution at $z\approx2-3$. In particular, the LAEs have extremely
high [OIII] $\lambda$5008/H$\beta$ ratios (log([OIII]/H$\beta$) $\sim$ 0.8) and
low [NII] $\lambda$6585/H$\alpha$ ratios (log([NII]/H$\alpha$) $<-1.15$). Using
the [OIII] $\lambda$4364 auroral line, we find that the star-forming regions in
faint LAEs are characterized by high electron temperatures
($T_e\approx1.8\times10^4$K), low oxygen abundances (12 + log(O/H) $\approx$
8.04, $Z_{neb}\approx0.22Z_\odot$), and high excitations with respect to more
luminous galaxies. Our faintest LAEs have line ratios consistent with even
lower metallicities, including six with 12 + log(O/H) $\approx$ 6.9$-$7.4
($Z_{neb}\approx0.02-0.05Z_\odot$). We interpret these observations in light of
new models of stellar evolution (including binary interactions). We find that
strong, hard ionizing continua are required to reproduce our observed line
ratios, suggesting that faint galaxies are efficient producers of ionizing
photons and important analogs of reionization-era galaxies. Furthermore, we
investigate physical trends accompanying Ly$\alpha$ emission across the largest
current sample of combined Ly$\alpha$ and rest-optical galaxy spectroscopy,
including 60 faint LAEs and 368 more luminous galaxies at similar redshifts. We
find that Ly$\alpha$ emission is strongly correlated with nebular excitation
and ionization and weakly correlated with dust attenuation, suggesting that
metallicity plays a strong role in determining the observed properties of these
galaxies by modulating their stellar spectra, nebular excitation, and dust
content.
