I'm just starting to learn about conformal field theory, with an aim to understand critical exponents in terms of conformal fixed points. I see, in multiple locations, the claim that a central charge $c<1$, $c=1-\frac{6}{m(m+1)}$ for $m=3,4,5,...$, dictates the universality class. However, I also see claims of the complete opposite! Because I am new to this field, I will share examples of conflicting claims, bolding what I view as the important parts, in case I am misunderstanding something fundamental.

For example, In 1d criticality, what is the relation between the universality class and central charge? says that "I want to know how to obtain the universality class of the phase transition from the central charge "c" in one dimensional model. If c is less than 1, there is a one-to-one correspondence." The paper 1D Fermi Liquids by Johannes Voit notes that "The critical exponents are the scaling dimensions of the various operators in a conformally invariant theory and, generically, are fully determined by c."

However, on nLab, there's the following quote of Cardy's:

"Shortly thereafter Friedan, Qiu and Shenker showed that unitary CFTs (corresponding to local, positive definite Boltzmann weights) are a subset of this list, with $c=1−\frac{6}{m(m+1)}$ and $m$ an integer $\geq 3$. This gives rise to what might be termed the "conformal periodic table". The first few examples may be identified with well-known universality classes. The "hydrogen atom" of CFT is the scaling limit of the critical Ising model, "helium" is the tricritical Ising model, and so on. Note, however, that at the next value of $c=4/5$ two possible "isotopes" arise. In the second, corresponding to the critical 3-state Potts model, not all the scaling dimensions allowed by BPZ in fact occur, but some of those that do actually appear twice. In fact the constraint of unitarity is not sufficient to determine exactly which representations actually occur in a given CFT."

Cardy's quote seems to say that $c<1$ does not fix the universality class, as there are two different CFTs for $m=5$.

Is it correct that $c<1$ does not fix the universality class? If it doesn't, is it known how many universality classes there are for a given $c<1$?


1 Answer 1


Cardy's statement is correct, as you can see by the explicit $c=4/5$ case. In this case, there are two different modular-invariant choices for the field content, one of which corresponds to the three-state Potts model and the other of which describes a generic tetra-critical point.

Once you have the central charge, you can write down the allowed scaling dimensions of all primary operators, but this isn't the end of the story. It turns out that modular invariance constrains which of these operators can appear in your theory. A complete classification of the field content allowed in the Virasoro minimal models is known as the ADE classification: http://www.scholarpedia.org/article/A-D-E_Classification_of_Conformal_Field_Theories . Directly quoting the linked article by Cappelli and Zuber:

The results in Table 3 amount to a classification of the operator contents of all rational conformal theories with c<1 ; we see that more than one consistent set of primary fields is possible for the same central charge (14), leading to quite different theories. These correspond to independent universality classes of critical phenomena in statistical mechanics, because the (relevant) primary fields characterize the manifold of perturbations around the fixed point.

  • $\begingroup$ I feel on more solid ground, reading this. I'll read up on the ADE classification after getting my bearings, thanks for the link. $\endgroup$
    – user196574
    Jul 11, 2021 at 2:34

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