We are presenting a model of field emission from carbon nanostructures. The underlying assumption of the model is the existence of a small-radius polaron in the emission center. Our research demonstrates that the polaron does not leave the emission center. The model explains the low work functions and the high cathode temperatures calculated using the emission characteristics. The unique mixture of sp2 and sp3 hybridized carbon atoms coexisting in the emission center was reflected in a formalized manner in the high value of the Migdal parameter for the emission center material. Based on such materials, it is possible to create new materials with high-T superconductivity.