Words and Thoughts: Monstrous Tits, From Belgium
Hello again, alleged readers! And boy howdy, do I hope you are in fact readers, because does Ol’ Stav have some stuff, in the classical sense, for y’all to read.
It all started in the office on Monday. I had my whole calendar blocked to research moonshine, because I’m pretty sure I could make it in my neighbor’s yard. Front or back, either yard would work really. His property just has a certain lawn service, picket fence, and buntings je ne sais quoi that could benefit from a moonshine still, in my opinion.
Anyway, right around 2pm I came across something called “monstrous moonshine”. Thinking I stumbled on some exciting variety of moonshine I was elated. Imagine my immediate dejection when I discovered that monstrous moonshine is not in fact moonshine at all. It’s math. As you know, I recently learned math, so despite my disappointment that monstrous moonshine was not potentially what I’d be brewing in the next yard over, I thought it’d be a nice break from the boozy browsing I’d been doing all day. And was it ever! And now I’ll share what I learned with all of you!
Back in the 70’s Belgian mathematician Jacques Tits gave a lecture and some other math people were there. Everyone knows Jacques Tits because he invented the Tits alternative, Tits building, Tits cone, and Tits group. But did you also know he invented the Tits index, Tits metric, and was involved in the Bruhat – Tits fixed point theorem?? So yeah, Tits was giving a lecture, and he may have actually renounced his Belgian citizenship by this time, but that’s neither here nor there. Tits was giving a lecture to a bunch of other math people who had been studying the quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of the Hecke congruence subgroup Γ0(p) in SL(2,R), as anyone would really, it was the 70s. So these math guys are all just conjecturing back and forth about the alleged finding that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ0(p)+ has genus zero exactly for p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. Little did they know at the time (how could they have?) that this would form the basis of “monstrous moonshine”, aka the unexpected connection between the monster group M and modular functions, in particular the j function.
I mean, we all know what the monster group is, but who would have guessed that connection between the group M and the modular functions existed? If we had known at the time about the underlying vertex operator algebra, maybe we’d have a clue, but back then everyone had no idea! It wasn’t until the early 90’s that Goddard and Thorn’s “No Ghost Theorem” that describes properties of a functor that quantizes bosonic strings could even prove the original conjecture. Of course, the “ghosts” from the Pauli and Villars regularization are a divergence arising from a loop integral modulated by a spectrum of auxiliary particles added to the Lagrangian propagator. And yet when we do it without ghosts, the absurd connection between the countably infinite families of the monster group and the holomorphic function on the complex upper half-plane H that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition becomes apparent!
OK, back to the actual moonshine. I’ll fax you when it’s ready.
To keep things simple for everyone, and to do the bare minimum to dodge any misguided accusations of plagiarism, the preceding fax is comprised of a selection of direct quotes, and a sprinkling of editing for clarity by yours truly, from the Wikipedia Pages for Jaques Tits, Monstrous Moonshine, Monster Group, Modular Form, Goddard-Thorn Theorem, and Pauli-Villars regularization.
Works Cited
Wikipedia Contributors. “Goddard–Thorn Theorem.” Wikipedia, Wikimedia Foundation, 18 Dec. 2025, en.wikipedia.org/wiki/Goddard%E2%80%93Thorn_theorem. Accessed 20 Apr. 2026.
—. “Jacques Tits.” Wikipedia, Wikimedia Foundation, 11 Mar. 2026, en.wikipedia.org/wiki/Jacques_Tits. Accessed 20 Apr. 2026.
—. “Modular Form.” Wikipedia, Wikimedia Foundation, 14 Aug. 2022, en.wikipedia.org/wiki/Modular_form. Accessed 20 Apr. 2026.
—. “Monster Group.” Wikipedia, Wikimedia Foundation, 19 Apr. 2026, en.wikipedia.org/wiki/Monster_group. Accessed 20 Apr. 2026.
—. “Monstrous Moonshine.” Wikipedia, Wikimedia Foundation, 8 Apr. 2026, en.wikipedia.org/wiki/Monstrous_moonshine. Accessed 20 Apr. 2026.
—. “Pauli–Villars Regularization.” Wikipedia, Wikimedia Foundation, 18 Nov. 2025, en.wikipedia.org/wiki/Pauli%E2%80%93Villars_regularization. Accessed 20 Apr. 2026.
