We study Kähler-Dirac fermions on Euclidean dynamical triangulations. This fermion formulation furnishes a natural extension of staggered fermions to random geometries without requiring vielbeins and spin connections. We work in the quenched approximation where the geometry is allowed to fluctuate but there is no backreaction from the matter on the geometry. By examining the eigenvalue spectrum and the masses of scalar mesons we find evidence for a fourfold degeneracy in the fermion spectrum in the large-volume, continuum limit. It is natural to associate this degeneracy with the well-known equivalence in continuum flat space between the Kähler-Dirac fermion and four copies of a Dirac fermion. Lattice effects then lift this degeneracy in a manner similar to staggered fermions on regular lattices. The evidence that these discretization effects vanish in the continuum limit suggests both that lattice continuum Kähler-Dirac fermions are recovered at that point, and that this limit truly corresponds to smooth continuum geometries. One additional advantage of the Kähler-Dirac action is that it respects an exact U(1) symmetry on any random triangulation. This U(1) symmetry is related to continuum chiral symmetry. By examining fermion bilinear condensates we find strong evidence that this U(1) symmetry is not spontaneously broken in the model at order the Planck scale. This is a necessary requirement if models based on dynamical triangulations are to provide a valid ultraviolet-complete formulation of quantum gravity.
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)