What is a "killer-app" for the formalism of topological quantum field theory in "established real world physics"?
To be more precise, I'm looking for an actual physical experiment, state of matter or something alike, together with a article that well describes it by a topological quantum field theory.
I'm, in particular, not looking for hypothesized physics like string theory, just for something that already has been observed.
Answer
There is a nice pedagogical review of the quantum Hall effect that can be found here. They explain (sketchily) how to derive an effective action describing the bulk of a quantum Hall fluid, which is a topological quantum field theory: Chern-Simons (CS) theory. The main physics which can be gleaned from this is that any defects in the bulk of the fluid could be anyonic quasiparticles. However, to my knowledge these defects are not observed in experiments; instead, the anyonic quasiparticles live on the edges of the sample. So this is hardly a "killer application", actually the CS theory just tells you that nothing interesting really happens in the bulk.
Of course, this review is pretty old and I expect that the state-of-the-art has changed considerably since then.
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