  We investigate whether quantum gravity fluctuations can break chiral symmetry
for fermions that are charged under a $U(1)$ and an $SU(N_\text{c})$ gauge
symmetry and thus closely resemble Standard-Model fermions. Unbroken chiral
symmetry in the quantum-gravity regime is a necessary prerequisite to recover
the Standard Model from a joint gravity-matter theory; if chiral symmetry is
broken by quantum gravity, fermions cannot generically be much lighter than the
Planck mass and the theory is ruled out. To answer this, we work in a
Fierz-complete basis of four-fermion interactions and explore whether they are
driven to criticality.
  We discover that the interplay of quantum gravity with the non-Abelian gauge
theory results in chiral symmetry breaking, because gravitational and gauge
field fluctuations act together to produce bound states.
  Chiral symmetry breaking is triggered by four-fermion channels that first
appear when non-Abelian charges are introduced and that become critical if the
non-Abelian symmetry is gauged. Extrapolating our result to the Standard Model
fermions, we conjecture that the non-Abelian gauge coupling, Abelian gauge
coupling and Newton coupling are all bounded from above, if Standard Model
fermions are to remain much lighter than the Planck mass.
  In contrast, fermions that are charged under a global non-Abelian group can
remain light for arbitrarily large values of the Newton coupling. We find that
different chiral symmetries are emergent at low energies, depending on the
strength of the gravitational coupling. This is an example of fixed points with
different degrees of enhanced global symmetry trading stability in fixed-point
collisions driven by gravitational fluctuations.
