The 511 keV γ-ray emission from the galactic center region may fully or partially originate from the annihilation of positrons from dark matter particles with electrons from the interstellar medium. Alternatively, the positrons could be created by astrophysical sources, involving exclusively standard model physics. We describe here a new concept for a 511 keV mission called 511-CAM (511 keV gamma-ray camera using microcalorimeters) that combines focusing γ-ray optics with a stack of transition edge sensor microcalorimeter arrays in the focal plane. The 511-CAM detector assembly has a projected 511 keV energy resolution of 390 eV full width half maximum or better, and improves by a factor of at least 11 on the performance of state-of-the-art Ge-based Compton telescopes. Combining this unprecedented energy resolution with sub-arcmin angular resolutions afforded by Laue lens or channeling optics could make substantial contributions toward identifying the origin of the 511 keV emission through discovering and characterizing point sources and measuring line-of-sight velocities of the emitting plasmas.