# Mysterious relationship between central charges of conformal field theories and the Beraha numbers

Background:

• Conformal field theories (CFTs) in two dimensions are partially characterized by a so-called central charge (characterizing the central extension of the Virasoro algebra which defines it). Under a condition of unitarity and minimality, all CFTs with $c<1$ have been classified. These make up the so-called minimal models $\mathcal M(m+1,m)$, where $m$ can be any integer $\geq 2$. The central charge $c_m$ of $\mathcal M(m+1,m)$ is given by $$c_m = 1 - \frac{6}{m(m+1)}$$
• Seemingly unrelated, there is the notion of the Beraha numbers. These are important numbers in algebraic graph theory, appearing in the study of the roots of so-called chromatic polynomials, which count the ways you can color graphs with a given set of colors. They are given by $$B_m = 4\cos^2 \left( \frac{\pi}{m} \right)$$

I am somewhat struck by a curious relationship between the central charge $c_m$ of the minimal model $\mathcal M(m+1,m)$ and the Beraha number $B_{m+1}$. To let the numbers speak for themselves:

$\begin{array}{c|c|c} m & 4^{c_m} & B_{m+1} \\ \hline 2 & 1 & 1 \\ 3 & 2 & 2 \\ 4 & 4^\frac{7}{10} \approx 2.639 & 1+\textrm{golden ratio} \approx 2.618 \\ 5 & 4^\frac{4}{5} \approx 3.031 & 3 \\ 6 & 4^\frac{6}{7} \approx 3.281 & \textrm{silver constant} \approx 3.247\\ 7 & 4^\frac{25}{28} \approx 3.448 & 2 + \sqrt{2} \approx 3.414\\ 8 & 4^\frac{11}{12} \approx 3.564 & 2+2\cos\left(\frac{2\pi}{9}\right)\approx 3.532 \\ 9 & 4^\frac{14}{15} \approx 3.647 & \frac{1}{2} \left( 5+\sqrt{5} \right) \approx 3.618 \\ 10 & 4^\frac{52}{55} \approx 3.709 & 2+2\cos\left(\frac{2\pi}{11}\right)\approx 3.683 \\ 11 & 4^\frac{21}{22} \approx 3.756 & 2+\sqrt{3} \approx 3.732 \\ 12 & 4^\frac{25}{26} \approx 3.792 & 2+2\cos\left(\frac{2\pi}{13}\right) \approx 3.771 \\ \vdots & \vdots & \vdots \\ \infty & 4 & 4 \end{array}$

As $m$ increases, the relationship gets obscured (or to phrase it more pessimistically, becomes perhaps less meaningful) since all the values get bunched up near each other (although it is curious to note that always $4^{c_m} \geq B_{m+1}$). Nevertheless the above table contains various apparently serendipitous relationships, suggesting --at least to me-- that there must be some conceptual link between $4^{c_m}$ and $B_{m+1}$. One possible relationship, might be if $4^{c_{m+1}}$ occurs as the zero of a chromatic polynomial, since the Beraha numbers are known to be accumulation points of such zeros (in the infinite graph limit).

How to explain' the above table?

To be clear, both numbers $c_m$ and $B_{m+1}$ have been known to appear together, e.g. in the discussion of Potts models (of which my knowledge is limited), but no direct relationship (of the type that I am hinting at) has been discussed.

EDIT: The inequality $4^{c_m} \geq B_{m+1}$ can be given a physical interpretation and justification. This is described in a recent manuscript of ours, but I will sketch the idea here. It is clearer in the equivalent form $c_m \geq \log_2 d_m$ where $d_m = \sqrt{B_{m+1}}$. The minimal model $\mathcal M(m+1,m)$ describes the phase transition between a so-called trivial phase and a topological' phase of quantum matter (in one spatial dimension), the latter exhibiting topological edge modes with quantum dimension $d_m$. Since the transition is exactly described by delocalized edge modes, and the central charge is known to be proportional to the massless degrees of freedom, one expects a relationship between $c$ and $d$. This has lead to the conjecture $c \geq \log_2 d$ (for the general case of a transition between a trivial and a topological phase) of which this is but a special case. That being said, it does not explain why this lower bound on the central chargea is almost-but-not-quite saturated. Indeed, this is how we noticed this funny relationship between $c_m$ and $B_{m+1}$.

• This pair of papers by Fendley and Krushkal might be related: arxiv.org/abs/0711.0016 arxiv.org/abs/0806.3484 . – j.c. Jul 19 '17 at 0:22
• It could be related to: mathoverflow.net/q/70575/34538 – Sebastien Palcoux Jul 19 '17 at 11:59
• There is nothing deep or mysterious here, $c_m$ and $\mathrm{log}_4 B_m$ just have approximately equal Taylor expansions in $1/m$. Honestly, both functions are so elementary that some relation between them isn't surprising. – Anton Fetisov Jul 20 '17 at 15:59
• @AntonFetisov Why should there be a relationship at all? I agree that if I had sampled from the space of all functions and I searched for (and found) two seemingly different functions that have vaguely similar values, of course that would not be surprising. In this case, however, as evidenced by the comments by Sebastian Palcoux and j.c. (thanks!), there are clearly deep links between the two topics in which these numbers appear. The fact that their numerical values subsequently turn out to be linked by a seemingly haphazard exponentiation involving $c_m$ makes a link all the more tantalizing. – Ruben Verresen Jul 20 '17 at 16:14
• (note that my 'EDIT' above moreover indicates that the exponentiation involved is in fact not as random as it might seem on first sight, at least from a physical perspective) – Ruben Verresen Jul 20 '17 at 16:16