  A common framework of particle physics consists of two sectors of particles,
such as the Standard Model and a dark sector, with some interaction between
them. In this work, we initiate the study of a qualitatively different setup in
which one of the sectors is a topological quantum field theory (TQFT). Instead
of particles, the physics of a TQFT only manifests itself in non-trivial
spacetime topologies. Topological defects provide a natural place to
investigate such effects. In particular, we consider two possible ways in which
axionic cosmic strings can interact with a Zn TQFT. One of them, by extending
the structure of the axion coupling, leads to specific predictions for the
localized degrees of freedom on the cosmic string, which can in turn effect
their evolution and leave observable signals. The second approach, by gauging a
discrete subgroup of the axionic shift symmetry, leads to dramatic changes in
the string spectrum. We stress that the scenario considered here should be
regarded as a plausible way for new physics to arise since it can be the low
energy effective field theory for quite generic scenarios at high energies. To
demonstrate this point and further illustrate the physical implications, we
constructed such UV completions for both of the cases of couplings to TQFTs.
The detailed prediction for observable signals of such scenarios needs further
investigation. At the same time, our results demonstrate that there are rich
new phenomena in this scenario.
