The classification of the "minimal models" of chiral algebra gives us rational conformal field theories in two dimensions. For example, the classification of unitary representations of Virasoro algebra gives us minimal models, which have $Z_2$ global symmetry. Similarly, the minimal models of $W_3$ algebra all have $S_3$ symmetry, the leading member of this family corresponds to the critical 3-state Potts model.

Suppose I construct a statistical physics lattice model in two dimensions, which preserves a certain finite group symmetry, one natural question is whether this lattice model, in the thermodynamic limit, will go through a second-order phase transition as we change temperature. (It is not difficult to construct such lattice models, for example, one introduces a spin degree of freedom on the site of a square lattice, which transforms in a certain irrep of the finite group, one can then use the invariant tensor of this finite group to introduce coupling among neighboring spins).

To answer this question, it seems to me that the first question to ask is whether there is a 2d CFT whose fusion rule preserves such a finite group symmetry. This lead to the question in the title "Given a finite group, how to figure out which chiral algebra can realize this symmetry?" (Will coset construction give us some hint?)


1 Answer 1


The global symmetries you cite are properties of specific models, not of chiral algebras. With a Virasoro algebra, you could have no global symmetry (Lee-Yang minimal model) or $S_Q$ ($Q$-state Potts model) or $O(N)$, etc.

Even if your chiral algebra has a nontrivial group of automorphism, for example $S_N$ for an algebra made of $N$ copies of Virasoro, this could in principle be broken in a specific model.

If you start with the $2d$ Ising model on the lattice, you notice the $\mathbb{Z}_2$ symmetry, then you argue that the critical limit exists, and then you start looking for a CFT with Virasoro and $\mathbb{Z}_2$ symmetries. The minimal model $M(4, 3)$ has the required symmetries, but it may not be what you want: if you are interested in nonlocal objects such as cluster connectivities, you need a bigger CFT.

  • $\begingroup$ Thank you for the answer, Sylvian! I understand now that the flavor symmetry groups should be associated with the full OPE instead of just the chiral algebra. I still want to understand "Given a finite group, how to construct a 2d CFT whose OPE respects such a symmetry?" $\endgroup$ Jan 2, 2023 at 15:34
  • $\begingroup$ Actually, I have a more specific following up question. Many rational CFTs have a coset construction, is there a relationship between the flavor symmetry respected by the fusion rules and the coset structure? $\endgroup$ Jan 2, 2023 at 15:36
  • $\begingroup$ Your first question seems too broad. We could decompose it in two parts: find a fusion category that respects the symmetry, and find a CFT with that fusion category. Both parts could be asked as independent questions. The formalism of Fuchs, Runkel and Schweigert might help. arxiv.org/abs/hep-th/0204148 $\endgroup$ Jan 3, 2023 at 8:16
  • $\begingroup$ For your second question, the answer must be yes there is a relationship, the real question is which relationship? This would be a good independent question, especially if you can give one or two simple examples before asking it in general. $\endgroup$ Jan 3, 2023 at 8:18

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