Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological quantum computation. Such as Ribbon Categories, Modular functors, etc Does anyone have other recommendations, for fusion algebras, the underlying representation theory,and quantum topology. Are Kevin Walkers TQFT notes good for instance? I understand a lot of Chern Simons theory, but still have problems and most of CFT is alien to me. So what are some good sources for the Mathematics of the Fractional Quantum Hall effect and Topological Quantum Computation. I appreciate however that the vast majority of the literature from a Mathematical point of view is in Topological Quantum Field Theories. Audience: I'm a Mathematics Masters student with a BSc in Physics, so can stomach a fair bit of quantum mechanics. I do however find the literature on CFT to be intimitating!
Here is a review of FQHE for mathematicians.
The fractional quantum Hall effect contains much semiconductor physics that will likely only distract you, as a mathematician. In particular, it is not even established whether the fractional quantum Hall effect supports a topologically nontrivial phase at all. (Much of the uncertainy comes from the fact that the two-dimensional electron gas actually extends in the third dimension.) From a mathematical perspective, this link contains a variety of pointers to the literature:
http://www.math.ucsb.edu/~jliptrap/ is a link to some recent work in Quantum Topology, at UCSB by a PhD student of Zhenghan Wang. There is some mathematics of the fractional quantum hall effect there.