Our campaign of deep monitoring observations with {it Chandra} of the nearby elliptical galaxy NGC 3379 has lead to the detection of nine globular cluster (GC) and 53 field low mass X-ray binaries (LMXBs) in the joint {it Hubble}/{it Chandra} field of view of this galaxy. Comparing these populations, we find a highly significant lack of GC LMXBs at the low (0.3-8 keV) X-ray luminosities (in the $sim 10^{36}$ to $sim 4times10^{37}$ erg s$^{-1}$ range) probed with our observations. This result conflicts with the proposition that all LMXBs are formed in GCs. This lack of low-luminosity sources in GCs is consistent with continuous LMXB formation due to stellar interactions and with the transition from persistent to transient LMXBs. The observed cut-off X-ray luminosity favors a predominance of LMXBs with main-sequence donors instead of ultra-compact binaries with white-dwarf donors; ultra-compacts could contribute significantly only if their disks are not affected by X-ray irradiation. Our results suggest that current theories of magnetic stellar wind braking may work rather better for the unevolved companions of GC LMXBs than for field LMXBs and cataclysmic variables in the Galaxy, where these companions may be somewhat evolved.
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