The system is topological; so what? Lately I've been studying topological quantum field theory, mainly from mathematically oriented sources and, while I understand why a mathematician would care about TQFT's, I'm not yet quite sure why a physicist would.
In particular, I'd like to know how TQFT's are used in particle physics. Say I begin with a standard QFT consisting of some gauge and matter fields. Somehow, I'm able to argue that in the infra-red, the theory flows to a Chern-Simons theory with gauge group $G$ and level $k$. Now what? What do I learn about the system once I realise its low-energy behaviour is described by a TQFT? What measurables can I study? What "real-life" predictions can I extract from the system?
Note: here I am only interested in (relativistic) particle physics. I know that TQFT's are also used in condensed matter and quantum gravity, but that's not what I want to understand here.
 A: Any gapped field theory flows in the infrared to a TQFT which describes the set of operators which are neither screened nor confined. See this paper for a very clear point of view on this: https://arxiv.org/abs/1307.4793 .
In gapped gauge theories like QCD, this is equivalent to specifying the dyon content of the theory ( https://arxiv.org/abs/1305.0318 ). Different possible dyon charge lattices correspond to different topological phases for the center 1-form symmetry ( https://arxiv.org/abs/1308.2926 ).
This can cause some unexpected things to happen. For instance, as one tunes the theta angle, at special values there are topological phase transitions between confining vacua with distinct dyon charge lattices ( https://arxiv.org/abs/1703.00501 ) and this accounts for a singularity in the axion potential, which has consequences for the strong CP problem ( https://arxiv.org/abs/1709.00731 ).
Here is another nice paper about line operators in the Standard Model, which discusses the phenomenological issues quite clearly, in an attempt to first answer what is the center of the SM gauge group, eg. whether it's simply-connected or not ( https://arxiv.org/abs/1705.01853 ). One has to answer this question before asking the ones above about the topological phase of the associated 1-form symmetries.
To put it all from a more abstract perspective, there are no known unitary 3+1D TQFTs which are not some kind of twisted gauge theory. We know some non-abelian ones, but there are no-go results for realizing them in familiar systems of SU(n) gauge fields interacting with fermions ( https://arxiv.org/abs/1106.0004 ). Thus, all of the questions of topological order in the Standard Model are equivalent to questions about the strong CP problem and about the center of the gauge group.
But still we haven't seen a magnetic monopole...
