Moonshine over the integers

Secret Blogging Seminar 2024-08-27

I’d been meaning to write a plug for my paper A self-dual integral form for the moonshine module on this blog for almost 7 years, but never got around to it until now. It turns out that sometimes, if you wait long enough, someone else will do your work for you. In this case, I recently noticed that Lieven Le Bruyn wrote up a nice summary of the result in 2021. I thought I’d add a little history of my own interaction with the problem.

I first ran into this question when reading Borcherds and Ryba’s 1996 paper Modular Moonshine II during grad school around 2003 or 2004. Their paper gives a proof of Ryba’s modular moonshine conjecture for “small odd primes”, and it has an interesting partial treatment of the problem of finding a self-dual integral form of the monster vertex algebra with monster symmetry. More explicitly, the authors wanted the following data:

An abelian group $V_\mathbb{Z}$ graded by non-negative integers, with finitely generated free pieces in each degree.
A multiplication structure $V_\mathbb{Z} \otimes V_\mathbb{Z} \to V_\mathbb{Z} ((z))$ satisfying the vertex algebra axioms over the integers.
A faithful monster action by vertex algebra automorphisms.
An integer-valued inner product that is self-dual (i.e., it gives a unimodular lattice for each graded piece), monster-invariant, and invariant in a vertex-algebraic sense.
A vertex algebra isomorphism $V_\mathbb{Z} \otimes \mathbb{C} \to V^\natural$ from the complexification to the usual monster vertex algebra. This is the “integral form” property.

These properties would allow for the following developments:

The monster action would let them consider the action of centralizers on fixed-point subalgebras.
The self-dual form gives an isomorphism between degree 0 Tate cohomology for a prime order element and the quotient of fixed points by the radical of the induced bilinear form. This connects the Tate cohomology formulation in Borcherds-Ryba to the original formulation of Modular moonshine given by Ryba.
The integral form property lets them connect mod p data to the traces of elements on $V^\natural$ , which are known from monstrous moonshine.

Unfortunately, the technology for constructing such an object was missing at the time, so a large fraction of their paper was spent making some partial progress on this problem and working around the parts they couldn’t finish. As it happens, the first progress toward an integral form was earlier, in the 1988 book by Frenkel, Lepowsky, and Meurman where they constructed $V^\natural$ . After the initial construction, near the end of the book, they exhibited a monster-symmetric form over the rationals. Borcherds and Ryba showed that this construction could be refined to work over $\mathbb{Z}[1/2]$ , and they gave some tantalizing hints for refining this to an integral form. In particular, they pointed out that we can make a self-dual integral form from self-dual forms over $\mathbb{Z}[1/2]$ and $\mathbb{Z}[1/3]$ , if they are isomorphic over $\mathbb{Z}[1/6]$ . In algebraic geometry language, this is “descent by a Zariski cover”.

Unfortunately, it seems to be quite difficult to construct a self-dual integral form over $\mathbb{Z}[1/3]$ . The construction of $V^\natural$ by Frenkel, Lepowsky, and Meurman starts with the Leech lattice vertex algebra (which has an “easily constructed” self-dual integral form), and applies eigenspace decompositions for involutions in an essential way. In general, if you do a construction using eigenspace decomposition for a finite-order automorphism of a lattice, then you destroy self-duality over any ring where that order is not invertible. Recovering a self-dual object tends to require a lot of work by hand (e.g., adding a specific collection of cosets), which is impractical in an infinite dimensional structure.

Instead of the order 2 orbifold construction of Frenkel, Lepowsky, and Meurman, one can try an order 3 orbifold construction. Given such a construction, one can hope that it can be done over $\mathbb{Z}[1/3, \zeta_3]$ (now we know this is possible), and Borcherds and Ryba suggested a strategy for refining this to $\mathbb{Z}[1/3]$ (I still don’t know how to make their method work). Dong and Mason had tried to do an explicit order 3 orbifold construction in 1994, but after a massive calculation, they had to give up. The order 3 construction was eventually done in 2016 by Chen, Lam, and Shimakura using some newer technology in the form of pure existence theorems (in particular, using the regularity of fixed-point vertex subalgebras I proved with Miyamoto, and Huang’s modular tensor theorem). However, it was not clear how to do this construction over smaller rings.

Anyway, I had talked about the problem of constructing a self-dual integral form with Borcherds during grad school after reading his modular moonshine papers, and he mentioned that he had considered giving it to a grad student to figure out, but that it seemed “way too hard for a Ph.D. thesis”. After that, I just kept the problem in the fridge. Every so often, some new advance would come, and I would think about whether it would help with this question, and the answer would be “no”. Even after the existence of cyclic orbifolds over the complex numbers was established (I blogged about it here), the question of defining them over small rings in a way that ensured self-duality and monster symmetry was a seemingly impenetrable challenge.

The event that changed my outlook was a conversation with Toshiyuki Abe at a conference in Osaka in 2016. He kindly explained a paper he was writing with C. Lam and H. Yamada, and in particular, a way to produce $V^\natural$ “inside” an order 2p orbifold of the Leech lattice vertex algebra. Basically, you can take two copies of the Leech lattice vertex algebra, related by order 2p cyclic orbifold duality, and use them to generate a larger structure that contains $V^\natural$ . This was the advance I needed, because (after an easy generalization from order 2p to order pq) it let me produce self-dual forms of $V^\natural$ over small rings like $\mathbb{Z}[1/pq, \zeta_{pq}]$ without doing any explicit work.

After this, the pieces slowly fell into place. Once I had self-dual forms over enough small rings, I could try to glue them together to get a form over the integers. Using some results on maximal subgroups of the monster, I was able to show that the results of gluing are unique up to isomorphism and carry monster symmetry. However, I found out that the fundamentals of gluing are tricky if you’re not so good at commutative algebra. Perhaps there is a lesson here about the advantages of finding good collaborators.

Gluing problems

Suppose you have a diagram of commutative rings $R_l \leftarrow R \to R_r$ , together with an $R_l$ -module $M_l$ and an $R_r$ -module $M_r$ .

Question: What data and properties do we need to have a uniquely defined $R$ -module $M$ such that $R_l \otimes_R M \cong M_l$ and $M \otimes_R R_r \cong M_r$ ?

One rather obvious necessary condition is that we need $M_l \otimes_R R_r \cong R_l \otimes_R M_r$ , since both sides would be $R_l \otimes_R M \otimes_R R_r$ with different choices of parentheses. However, this is not sufficient, unless the diagram $R_l \leftarrow R \to R_r$ satisfies some additional properties.

If we consider this from the point of view of algebraic geometry, we have a diagram of schemes $\text{Spec} R_l \to \text{Spec} R \leftarrow \text{Spec} R_r$ and quasicoherent sheaves on the sources of the arrows. We would like to have a quasicoherent sheaf on $\text{Spec} R$ that pulls back to the sheaves we had. Clearly, if the scheme maps are not jointly surjective, then the sheaf on $\text{Spec} R$ will not be uniquely determined, since any point not in the image can be the support of a skyscraper sheaf.

We come to our first sufficient condition: If we have a Zariski cover, namely the two maps are open immersions that are jointly surjective, then a choice of isomorphism $M_l \otimes_R R_r \cong R_l \otimes_R M_r$ yields an $R$ -module $M$ together with isomorphisms $R_l \otimes_R M \cong M_l$ and $M \otimes_R R_r \cong M_r$ , and these data are unique up to unique isomorphism.

The problem in my situation was that I needed to glue modules using some maps that were not open immersions. When I wrote the first version of my paper, I was under the mistaken impression that I could glue sheaves on étale covers the same way we glue sheaves on Zariski covers (i.e., that we don’t need to consider fiber products of open sets with themselves), and this led to some strange conclusions. In particular, I thought I had constructed 4 possibly distinct integral forms.

After a referee asked for a reference for my claim, I realized that it was false! Here is a counterexample: Take a scheme with two connected components $X = U \cup V$ , and define a 2-element étale cover given by arbitrary surjective étale maps to each component: $\{ U_1 \to U, U_2 \to V \}$ . The gluing data gives no information, since the intersection is empty, so we can’t in general descend a sheaf along the surjective étale maps.

I eventually found a different sufficient condition: If both maps in the diagram $R_l \leftarrow R \to R_r$ are faithfully flat, then then a choice of isomorphism $M_l \otimes_R R_r \cong R_l \otimes_R M_r$ yields an $R$ -module $M$ together with isomorphisms $R_l \otimes_R M \cong M_l$ and $M \otimes_R R_r \cong M_r$ , and these data are unique up to unique isomorphism. The next problem was writing a solid proof of this new claim, and this required several more iterations with a referee because I wasn’t very careful.

Anyway, I am very grateful for the persistence and careful reading of referee 2, who prevented me from releasing a sloppy piece of work.

About the journal

I had thought about submitting my paper to a top-ranked journal, but my friend John Duncan asked me to submit it to a special SIGMA issue on Moonshine that he was editing. SIGMA is a “diamond open-access” ArXiv overly journal, and this suited my ideological leanings. Also, I had recently gotten tenure, so putting things in high-ranked journals suddenly seemed less important.