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1. What the question is asking (in plain language)

The poster wants a catalogue of the rational conformal field theories that are obtained by orbifolding the
(SU(2)_k) Wess–Zumino–Witten (WZW) model (and its (SO(3)_k) cousin) by a finite normal subgroup (G\subset SU(2)).

For each such orbifold the student would like to know

  Desired information
a) concrete examples (e.g. (G= \mathbb Z_2,\;D_n,\;H_8)…) together with the modular (S)‑ and (T)-matrices and the fusion rules of the resulting anyons (the “quasiparticles’’ of the 2+1‑d bulk topological field theory).
b) the central charge (c) of the 1+1‑d CFT, and the ground‑state degeneracy on a spatial torus (i.e. the number of simple objects of the bulk TQFT).
c) if possible, a description of the bulk theory as a twisted quantum double (D^{\omega}(G’)) of some finite group (G’).

The answer must give the above data explicitly (or at least the general formulas that allow one to compute it) for the most common sub‑groups of (SU(2)) – cyclic, dihedral and the three binary polyhedral groups – and work out the case (G=D_2\simeq\mathbb Z_2\times\mathbb Z_2) (the quaternion group (H_8) is isomorphic to the binary dihedral group of order 8).

2. Orbifolding a WZW model – the general construction

2.1 The parent theory (SU(2)_k)

  • Chiral algebra: the affine Lie algebra (\widehat{\mathfrak{su}}(2)_k).
  • Central charge
    [ c\;=\;\frac{3k}{k+2}\; . \tag{2.1} ]
  • Primary fields (integrable highest‑weight representations) are labelled by a spin
    [ j = 0,\;\frac12,\;1,\;\dots,\;\frac{k}{2}. ] Their conformal weights are
    [ h_j = \frac{j(j+1)}{k+2}. \tag{2.2} ]

The theory possesses a global symmetry given by the centre of (SU(2)), [ Z(SU(2))={1,-\mathbf 1}\cong\mathbb Z_{2}, ] which acts on a primary of spin (j) by the phase ((-1)^{2j}). More generally any finite subgroup (G\subset SU(2)) (the binary polyhedral groups) acts as an automorphism of the chiral algebra – the action is simply the usual left multiplication on the group‑valued field (g(z,\bar z)) of the WZW model.

2.2 The orbifold theory ( \bigl(SU(2)_k\bigr)/G )

Given a finite group (G) of automorphisms, the orbifold is defined by

  1. Projection to (G)-invariant states (the untwisted sector).
  2. Adding twisted sectors labelled by conjugacy classes ([g]) of (G); a twisted sector is a copy of the original theory with boundary condition (g) around the spatial circle.
  3. Implementing the (G)-projection inside each twisted sector (the twist‑field projection).

The resulting chiral algebra is the (G)-invariant sub‑algebra of the original one, and the simple objects (primary fields) of the orbifold are in one‑to‑one correspondence with the pairs

[ \boxed{ \; (\,C,\;\rho\,) \;} \qquad \begin{array}{c} C \;=\; \text{conjugacy class of }G,\[2pt] \rho \;=\; \text{irreducible representation of the centraliser }N_g:={h\in G\mid hg=gh}, \end{array} \tag{2.3} ]

where we pick any representative (g\in C). This is precisely the label set of the (possibly twisted) quantum double (D^{\omega}(G)).

*If the original CFT is holomorphic (central charge a multiple of 8) the orbifold is *exact

Original question: Orbifold CFT of $SU(2)/G$ and $SO(3)/G$ on Physics Stack Exchange, licensed CC BY-SA.