Ly$alpha$ emission is a standard tracer of starburst galaxies at high redshift. However, a number of local Ly$alpha$ emitters (LAEs) are X-ray sources, suggesting a possible origin of Ly$alpha$ photons other than young, hot stars, and which may be active at much later ages relative to the parent starburst. Resolved, nearby LAEs offer the opportunity to discriminate between diffuse X-ray emission arising from supernova-heated gas, high-mass X-ray binaries (HMXBs), or low-luminosity active galactic nuclei (LLAGN). We examine archival X-ray imaging from Chandra and XMM-Newton for 11 galaxies with spatially resolved Ly$alpha$ imaging to determine the luminosity, morphology, and spectral hardness of the X-ray sources. The data are consistent with 9 of the 12, bright Ly$alpha$ sources being driven by luminous, $10^{40}$ erg s$^{-1}$ X-ray sources. Half of the 8 Chandra sources are unresolved. The data suggest that nuclear activity, whether from LLAGN or nuclear starbursts, may play an important role in Ly$alpha$ emission. Our results also suggest a significant link between Ly$alpha$ emission and HMXBs, ULXs, and/or LLAGN, which would imply that Ly$alpha$ may be generated over timescales 1 - 2 orders of magnitude longer than produced by photoionization from OB stars. This highlights a critical need to quantify the relative contributions of different sources across cosmic time, to interpret Ly$alpha$ observations and the resulting properties of distant galaxies.
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