# 71, the Monster, and c = 24 CFTs

The largest prime in the order of the Monster group is $71$. This number $71$ shows up at various places:

• The minimal faithful representation has dimension $196883 = 47.59.71$

• The Monster group can be realised as a Galois group $Gal$ $L(71)/{\mathbb{Q}(\sqrt{-71})}$ where $L(71)$ is a suitable field.

The appearance of $71$ in the above two cases (and possibly others) is not very surprising and might be reasoned out.

But the appearance of $71$ in a different area seems very intriguing:

The Monster is intimately connected to a special class of conformal field theories. These are the meromorphic $c = 24$ CFTs. The Monster here arises as the discrete automorphism group of the vertex operator algebra of one of the c =24 CFTs.

Schellekens in 1992 enumerated such CFTs and he found $71$ such CFTs! All these CFTs have a partition function of the form $$Z(\tau) = j(\tau) + \mathcal{N}$$ where $j$ is modular invariant and $\mathcal{N} \geq -744$ is an integer. But any value of $\mathcal{N}$ won't work. Schellekens found $71$ values of $\mathcal{N}$ which will work.

Unfortunately, it is still not clear if the enumeration Schellekens made is exhaustive, i.e. if there are only exactly $71$ such theories.

Is the appearance of $71$ here just a coincidence? Or is it again connected to the Monster? It is hard to believe that this is just a coincidence.

• As far as I know the 71 in Schelleken's classification is not related to the other occurrences you mention. Some recent work on the construction of Schelleken's examples can be found here arxiv.org/pdf/1708.05990.pdf – Jeff Harvey Sep 15 '17 at 19:44
• Schellekens remarks on this coincidence towards the end of his paper. But that was 25 years ago, and I wanted to know if there was any progress. @ Jeff Harvey The article is dedicated to Griess on the occasion of his 71st birthday, cool! It looks like a very useful article, thank you! – Ramesh Chandra Sep 15 '17 at 19:52
• One might also ask why, at magic number 24, there are n=24 self-dual even lattices of dimension 24. I know how most numbers 24 are related, but I don't yet know if |{lattices}| is the "same" 24 as all the others. – Theo Johnson-Freyd Sep 16 '17 at 3:29
• @TheoJohnson-Freyd For that matter, is it also related to the $576=24^2$-periodicity of TMF? – მამუკა ჯიბლაძე Sep 16 '17 at 4:29
• @მამუკაჯიბლაძე Here's how I want to explain the number 576. I will say it in condensed matter. As warm-up, Bott periodicity. If you allow interacting fermions, then there are 2 = \pi_2(stable sphere) invertible phases of (1+1)-dimensional matter. But there are 8 phases "protected by time-reversal symmetry" (which is the condensed matter way of saying "real"). Similarly, there should be 24 = \pi_3(stable sphere) invertible phases of (2+1)-dimensional matter. I expect this to expand to 576 when you protect by the appropriate symmetry. The symmetry group should be related to SL(2,Z)... – Theo Johnson-Freyd Sep 16 '17 at 4:39