Kähler-Dirac fermions on Euclidean dynamical triangulations

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.

Figure 1

Schematic of the phase diagram as a function of ${\kappa }_2$ and $\beta$.

Figure 2

A section of a lattice showing oriented 4-simplices. It is implied that at the end of each edge is another 4-simplex even if not drawn. Here the faces of adjacent 4-simplices must have opposite parity. Once the parity of a single face in a 4-simplex is known, all other parities for the faces in the 4-simplex are known (after fixing a vertex ordering). A spanning tree is shown in red.

Figure 3

The power-law behavior of the first few smallest eigenvalues of the Kähler-Dirac matrix in the system size volume. Here we show the volume dependence across the 4000, 8000, and 16 000 simplex ensembles at $\beta =0$. The dashed lines are linear fits across the three volumes.

Figure 4

The first 16 eigenvalues of the Kähler-Dirac matrix at infinite-volume extrapolation for the $\beta =0$ ensembles.

Figure 5

Meson propagators for the 3-simplex to 3-simplex block for four different masses, along with their fits in black using Eq. (37). Here the three different distance regions are shown (left to right) as yellow, green, and blue. This data is from the $\beta =-0.8$, 8000 simplex ensemble.

Figure 6

The chiral extrapolation for the 1-, 2-, and 3-simplex mesons. Here the dashed lines are linear fits, and the furthest left data points are the $m_0\to 0$ extrapolation. These masses are from the 4000 simplex, $\beta =0$ ensemble.

Figure 7

The physical masses of the meson propagators between like simplices in the chiral limit for a variety of ensembles. The 0- and 4-simplex masses tend to zero in the chiral limit because of the inherent zero modes. The other simplex mesons remain massive in the chiral limit; however, as the lattices become finer their masses tend to get smaller. The coarsest lattices do not follow this trend, but the expected scaling should break down for sufficiently large lattice spacing. The relative lattice spacing is set by the fiducial lattice spacing at $\beta =0$.

Figure 8

The sum of the squares of the meson masses for the 1-, 2-, and 3-simplex mesons versus the squared relative lattice spacing. At coarse lattice spacing there does not seem to be a particular trend; however, at finer lattices these masses trend to zero. The line is fixed at zero and the lower $a_{\mathrm{rel}}^2=1$ point and is meant to guide the eye.

Figure 9

The bilinear condensate for the Kähler-Dirac matrix at three different volumes and the extrapolation to infinite volume. The finite-volume curves show that the condensate diverges for sufficiently small values of $m_0$. Taking the infinite-volume limit and then taking the chiral limit we see that the condensate vanishes as $m_0\to 0$. The extrapolation to zero in the chiral limit indicates the $U(1)$ symmetry associated with Eq. (29) is not spontaneously broken. These calculations were done at $\beta =0$.