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!
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4$\begingroup$ Is your question "What are some references on the mathematics of FQHE or topological quantum computing?" It's not completely clear from the way you've phrased your post. This question is also a bit too broad in my view. Have you looked at e.g. the paper of Freedman et al in the Bulletin of the AMS: ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3 and how much of it did you follow? What kind of sources you'll want to read will be highly dependent on how much physics you know and have the stomach for. For instance, how's your quantum mechanics? $\endgroup$– j.c.Commented Sep 1, 2011 at 22:49
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$\begingroup$ Just to give some background. I've got a Bachelors in Physics, so I can stomach a lot of Quantum Mechanics. I shall edit my question. $\endgroup$– Peadar CoyleCommented Sep 2, 2011 at 9:33
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$\begingroup$ @Peadar: It looks like you got an acceptable answer below, so for this question my comment is probably moot. But can I ask that in the future you take a bit more care with punctuation? With @jc, I found your post a little hard to follow, and at least part of the problem for me is that I usually skim for question marks, but there are at least two sentences phrased as questions that end with periods. $\endgroup$– Theo Johnson-FreydCommented Sep 11, 2011 at 23:18
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$\begingroup$ Sorry for the confusion. Thank you for your comment. $\endgroup$– Peadar CoyleCommented Sep 25, 2011 at 15:19
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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:
answered Sep 1, 2011 at 20:59
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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.