Abstract
We study K\"ahler-Dirac fermions on Euclidean dynamical triangulations. This fermion formulation furnishes a natural extension of staggered fermions to random geometries without requring vielbeins and spin connections. We work in the quenched approximation where the geometry is allowed to fluctuate but there is no back-reaction from the matter on the geometry. By examining the eigenvalue spectrum and the masses of scalar mesons we find evidence for a four fold 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 K\"ahler-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 K\"ahler-Dirac fermions are recovered at that point, and that this limit truly corresponds to smooth continuum geometries. One additional advantage of the K\"ahler-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.
Highlights
Quantum gravity remains one of the outstanding challenges in theoretical physics
In Ref. [13], which involved a subset of the authors of this work, we presented evidence that Euclidean dynamical triangulations (EDT) supplemented by an ultralocal measure term [17] can recover semiclassical geometries that are four dimensional within measurement errors, and that this model has a continuum limit
We find evidence that these discretization effects vanish in the continuum limit, bolstering the case that we can include fermions with the expected continuum properties on dynamical triangulations, and perhaps more importantly, that a continuum limit exists at all
Summary
Quantum gravity remains one of the outstanding challenges in theoretical physics. One approach to obtaining a consistent, predictive theory is the asymptotic safety scenario of Weinberg, where the theory is effectively renormalizable when formulated nonperturbatively [1]. We do not want quantum gravity to lead to spontaneous chiral symmetry breaking induced at the Planck scale, since this would cause fermion bound states to acquire masses of order the Planck scale Notice that this is not just an academic point: early work with Dirac fermions on quenched random lattices saw precisely this phenomenon and one might worry that a similar phenomenon would occur on DT geometries [21,22]. In this work we study fermions in four dimensions using the Kähler-Dirac formulation [27] This is a natural approach to fermions on dynamically triangulated geometries because of the precise correspondence between the continuum and lattice formulations and because the construction does not require the addition of new degrees of freedom like the vielbein and spin connection.
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