Anyon and state spaces I start learning about anyons, but I'm confused by a few Hilbert spaces.
First of all, it is said that anyons are "excitations" with anyonic statistics. By that I would imagine they are states inside certain Hilbert space $\mathcal{H}_\text{original}$, and likely some states with lowest (excited) energies w.r.t. some Hamiltonian $H$.
However, I frequently encounter some objects that appear in different contexts (which I naively think are the same thing)

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*the "ground state degeneracy" of some mysterious "ground state Hilbert space $\mathcal{H}_1$" associated to some punctured Riemann surface, and this space seems to be topological, and when the Riemann surface is $T^2$,
$\dim \mathcal{H}_1 = \# $ anyons.


*the space $\mathcal{H}_\text{blocks}$ of conformal blocks of the boundary WZW model and its dimension;


*some mysterious Hilbert space $\mathcal{H}_2$ whose dimension is computed by a fusion chain, like $N_{ab}{^c}N_{cd}{^e} ...$;


*the mysterious "physical Hilbert space" $\mathcal{H}_\text{phy}$ discussed in section 3.4 in Witten's Jone's polynomial paper.
My questions would be:

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*Are $\mathcal{H}_1 = \mathcal{H}_\text{block} = \mathcal{H}_2 = \mathcal{H}_\text{phy}$?


*Is there relation between these $\mathcal{H}$'s and the original $\mathcal{H}_\text{original}$?


*What is the physical origin/intuitive understanding of these weird $\mathcal{H}$'s? Why is $\dim\mathcal{H}_1$ called "ground state degeneracy", if anyons are "excitations"?
 A: I'll try to address the questions about $\cal{H}_1, \cal{H}_2$ and $\cal{H}_\text{original}$.
It is important to distinguish a gapped topological phase which has anyonic excitations, v.s. a topological quantum field theory.
A gapped topological phase means that one considers a 2+1d Hamiltonian (lattice or continuum) with local interactions, with a spectral gap in the thermodynamic limit. This is a physical system, so there should be a Hilbert space, which you called $\mathcal{H}_\text{original}$. Then the low-lying excited states can be described in terms of weakly-coupled, localized excitations, which may obey anyonic statistics. These are the anyons.
If you put such a system on a Riemann surface (e.g. torus), there is a subspace of states below the gap. This is the ground state subspace. The dimension of this space on a torus is (often) equal to the number of distinct types of anyons. Note that here there are no punctures. These states are topological, in the sense that they are locally indistinguishable. This space is your $\mathcal{H}_1$. Because of the local indistinguishability, all these states have almost degenerate energy, so they are degenerate. The dimension of $\mathcal{H}_1$ is thus the ground state degeneracy. The reason that the dimension is equal to the number of anyon types requires a longer explanation.
We can also consider Riemann surface with punctures. Or perhaps more precisely, since we are now doing the Hamiltonian stuff, there are anyonic excitations pinned at certain locations (by locally modifying the Hamiltonian to trap these excitations). Then there is also a space of states which are all locally indistinguishable from each other. So we can also speak about the space of states associated with these fixed anyons. The dimension of this space can be computed using the fusion rules of the anyons, this is your $\mathcal{H}_2$.
Both $\mathcal{H}_1$ and $\mathcal{H}_2$ are low-energy subspace of $\mathcal{H}_\text{original}$, with respect to some Hamiltonian. They are both topological, in the sense that the states inside them are all locally indistinguishable.
Now all of this can also be phrased using a topological quantum field theory (TQFT). The TQFT does not start from a Hamiltonian. Rather, it is more like an axiomatic QFT that assigns a Hilbert space to each (closed and compact) spatial manifold. In most cases, the gapped system can be related to a TQFT, meaning that these spaces $\mathcal{H}_1$ and $\mathcal{H}_2$ are mathematically the same as the space assigned by the TQFT. However, in TQFT there is no notion of energy or excitation, and there are no local operators. So to talk about anyons one has to consider a manifold with punctures, with each puncture labeled by an anyon type.
