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Disclaimer. One might argue that my question is off topic as it is clearly a question about physics...
But I'd like a mathematically phrased answer, and I expect that only a mathematician can offer an answer that I would deem satisfying. I hope that this forum is the correct place to ask my question.

By the way, FQHE = Fractional quantum Hall effect

Consider a FQH fluid in topological phase, on a compact Riemann surface. That is a physical system and, presumably, there is a set of possible states of that physical system.

I'm guessing that that set comes with an equivalence relation: the relation of being physically undistinguishable. I'm not (only) asking about the set of equivalence classes under the above equivalence relation: I'm asking about the actual set of physical states.

Ok, this might sound perverse, so let me justify... I'm secretly guessing that the set of possible states might actually form the objects of a category (groupoid? higher category?), and that the above mentioned relation is that of isomorphism in the category. If that is indeed the case, then I change my question to: what is the category of possible physical states?

The next question (somewhat more relevant for actually physical applications), is the same one as above, but when the Riemann surface is allowed to have a boundary, e.g. when it is an open subset of ℂ.

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I'd suggest that you contact Greg Moore at Rutgers if you want an answer. He wrote an important paper on the FQHE (physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf) that mentions the topic you ask about, and is one of a smallish number of physicists with enough mathematical sophistication to give you an answer that you might deem satisfying. I don't think he is on MO. –  Jeff Harvey May 26 '12 at 16:42
to "set of possible states of that physical system" : That's just the Hilbert space of the quantum system. Furthermore there is no phase transition in FQHE / IQHE (in contrast to superconductivity). –  jjcale Oct 16 '13 at 19:42
@jjcale: I want an answer in terms of some effective theory, not an answer in terms of the electrons in the electron gas. Is there a way of describing your "Hilbert space of the quantum system" in terms of the effective theory only? (pick which ever model you want, e.g. the semion) –  André Henriques Oct 17 '13 at 21:42
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1 Answer

The wiki article on topological order may answer your question. The degenerate ground states of FQHE on closed Riemann surface and their modular transformation properties define the concept topological order. The space of the degenerate ground states is the vector space on the boundary in the theory of TQFT.

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Dear Prof. Wen, I understand that the ground states form a finite dimensional vector space, but I'm not asking about just ground the states. For example, I would guess that a possible state of the system is to have an anyon somewhere in the bulk of the Riemann surface. That anyon might then have its own internal degrees of freedom... My question is about the way to mathematically model that situation. –  André Henriques Oct 16 '13 at 13:07
The section II of my paper with Tian Lan discussed the situation with particles on Riemann surface: arxiv.org/abs/1311.1784 . In particular, we discussed the physical meaning of the notion of simple type and composite type. –  Xiao-Gang Wen Jan 19 at 9:51
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