Regarding question 2: FQHEs are examples of topological phases of matter, whose effective field theories are topological quantum field theories, whose quasi-particle excitations are (essentially) described by modular tensor categories (MTCs). Since what we're usually interested in for constructing models of topological quantum computing (and related things) is the algebraic data of these quasi-particles, the problem of classifying topological phases is very closely related to the problem of classifying modular tensor categories.
Since we're talking about categories what we really want to do is classify them up to something, which in this case is (braided) monoidal equivalence. It turns out though that, for every equivalence class of MTCs, representatives of that equivalence class can be constructed from solutions to certain polynomial equations called the pentagon, hexagon, and pivotal equations. The collection of these solutions define an algebraic set $X$.
For a given $X$ there exists an algebraic group $G$ which acts on $X$ such that for points $F\text{ and }F' \in X$, $F$ and $F'$ give rise to monoidally equivalent categories if and only if there exists $g \in G$ such that $g \cdot F = F'$. Thus the orbits of $G$ in $X$ are in 1-1 correspondence with equivalence classes of categories, and so now we can consider the problem of classifying orbits of $G$.
It turns out that in doing this we have almost the nicest possible situation imaginable - $G$ is reductive, all orbits have the same dimension and are in fact the irreducible components of $X$. This then implies that we can construct another algebraic set $Y$ which is an orbit space for $X$ - that is to say that the points of $Y$ are in 1-1 correspondence with orbits of $G$ in $X$ and the regular functions on $Y$ are those regular functions on $X$ which are invariant under the action of $G$. All of this allows us to classify orbits (i.e. MTCs) by looking at the evaluations of $G$-invariant functions on $X$.
Picking these functions is a really hard problem in general, but for MTCs we have a set of generic candidates: Every MTC gives you a pair of matrices $(S,T)$ which are the so called modular data of the category. They are called this because they specify a representation of the modular group $SL(2,\mathbb Z)$. It is conjectured that MTCs are classified by their modular data.
Bringing things back to physics, the $(S,T)$ matrices have physical meaning - the entries of the $S$-matrix encodes the mutual statistics between particle types and the $T$-matrix encodes the self-statistics. Additionally, the entries of $S$ and $T$ are given by the evaluations of regular functions on $Y$ which is to say that they are given by the evaluation of $G$-invariant regular functions on $X$.
That this has physical meaning can be seen by noting that $G$ is (essentially) the gauge group for our TQFT. Given two quasi-particles $a$ and $b$, the state space $V_{a b}$ for their composite system is finite dimensional and decomposes in to subspaces $V_{a b}^c$, where $c$ is another quasi-particle type (including the vacuum) and whose dimension is the number of fusion channels from $a\otimes b$ to $c$. $G$ is the direct product of the groups of basis transformations on the $V_{ab}^c$ spaces.
The information in paragraphs 1,6, and 7 is pretty standard and can basically be found in these lecture notes. The details for paragraph 2 can be found in arxiv:1305:2229 and for paragraphs 3 and 4 in arxiv:1509.03275.
This post imported from StackExchange MathOverflow at 2016-11-17 18:49 (UTC), posted by SE-user Matthew Titsworth
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