Spectral Dirac graphs

We address the problem of identifying families of discrete models naturally flowing in continuum limit to relativistic quantum field theories. We call them Dirac graphs. In this work, we require the graphs to obey spectrality property, which implies that the adjacency matrix can be represented by a continuous function defined on the spectral space. We also consider deformations of such graphs away from the spectrality. We show that deformations with regular continuum limit result in background gauge field and gravity. Interestingly, gauge interactions appear due to contribution microscopic deformations while the standard gravity can only be a result of macroscopic (adiabatic) deformations.