  We discuss the discretization of Yang-Mills theories on Dynamical
Triangulations in the compact formulation, with gauge fields living on the
links of the dual graph associated with the triangulation, and the numerical
investigation of the minimally coupled system by Monte Carlo simulations. We
provide, in particular, an explicit construction and implementation of the
Markov chain moves for 2D Causal Dynamical Triangulations coupled to either
$U(1)$ or $SU(2)$ gauge fields; the results of exploratory numerical
simulations on a toroidal geometry are also presented for both cases. We study
the critical behavior of gravity related observables, determining the
associated critical indices, which turn out to be independent of the bare gauge
coupling: we obtain in particular $\nu = 0.496(7)$ for the critical index
regulating the divergence of the correlation length of the volume profiles.
Gauge observables are also investigated, including holonomies (torelons) and,
for the $U(1)$ gauge theory, the winding number and the topological
susceptibility. An interesting result is that the critical slowing down of the
topological charge, which affects various lattice field theories in the
continuum limit, seems to be strongly suppressed (i.e., by orders of magnitude)
by the presence of a locally variable geometry: that may suggest possible ways
for improvement also in other contexts.
