  We explore the phenomenological consequences of breaking discrete global
symmetries in quantum gravity (QG). We extend a previous scenario where
discrete global symmetries are responsible for scalar dark matter (DM) and
domain walls (DWs), to the case of fermionic DM, considered as a feebly
interacting massive particle, which achieves the correct DM relic density via
the freeze-in mechanism. Due to the mixing between DM and the standard model
neutrinos, various indirect DM detection methods can be employed to constrain
the QG scale, the scale of freeze-in, and the reheating temperature
simultaneously. Since such QG symmetry breaking leads to DW annihilation, this
may generate the characteristic gravitational wave background, and hence
explain the recent observations of the gravitational wave spectrum by pulsar
timing arrays. This work therefore highlights a tantalizing possibility of
probing the effective scale of QG from observations.
