The (apparent, supposed) mathematics behind the fractional quantum hall effect involves the TQFT invariants coming from representations of quantum groups at roots of unity.

**Edit:** some links for further reading:

 + [From String Nets to Nonabelions](http://arxiv.org/abs/cond-mat/0610583)
 + [A class of P,T-invariant topological phases of interacting electrons ](http://dx.doi.org/10.1016/j.aop.2004.01.006)
 + [Topological quantum computation](http://dx.doi.org/10.1090/S0273-0979-02-00964-3)