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.
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.
Topological Quantum Field Theory (TQFT) is the low energy effective theory for the topological ordered states in real world, such as FQH states. In fact, the name "topological order" was motived from the term "Topological Quantum Field Theory". [See Topological Orders in Rigid States, Xiao-Gang Wen, Int. J. Mod. Phys. B4, 239 (1990) http://dao.mit.edu/~wen/pub/topo.pdf].
We also have a modern mathematical summary in arXiv:1405.5858 Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions which point out that:
Topological orders in $n$ space-time dimension = unitary $n$-categories with one object = $n$ dimensional fully extented TQFT = gravitational anormalies in one lower dimensions
Since topological orders describe gapped quantum phases in real world, $n$-categories, fully extented TQFT, gravitational anormalies all have connection to real word quantum states of matter, such as FQH state. The spins in 2+1D TQFT can be measured by edge tunneling IV curve or RT curve.