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.