Borcherds' Section 15 open problems on monstrous moonshine Lie (super)algebras: natural constructions, denominator identities, embeddings, and replicability

At the end of "Monstrous moonshine and monstrous Lie superalgebras" (1992), the paper in which he proved the Conway-Norton moonshine conjecture, Richard Borcherds listed eight open questions about the monster Lie algebra, its twisted and generalized siblings, and completely replicable functions. Later work settles three of those questions: the modular-form assumptions behind the twisted denominator formulas for the primes $p=2,3,5,7,11,23$ (established within the natural-construction works of Höhn-Scheithauer and Möller); the explicit vertex-algebraic description of the Lie bracket (Driscoll-Spittler, 2024; journal publication 2026); and Norton's genus-zero conjecture for non-degenerate completely replicable functions (Cummins-Gannon, 1997). The remaining core is open: natural (string-theoretic) constructions are still lacking for the majority of the twisted fake-monster Lie (super)algebras, including the required finite-group actions such as $2\cdot B$ on the baby monster Lie algebra; it is not known whether the baby monster Lie algebra embeds in the monster Lie algebra; no closed-form root multiplicities are known for the Lie algebras of root lattices $\mathrm{II}_{9,1}$ and $\mathrm{II}_{17,1}$; the denominator formulas of the Lie algebras and superalgebras coming from the remaining elements of the Monster or of $\operatorname{Aut}(\hat{\Lambda})$ are not written down explicitly in closed infinite-product form in general; and the search for further generalized Kac-Moody algebras with explicitly computable roots and multiplicities remains open-ended.

Background

Monstrous moonshine connects the Monster finite simple group $\mathbb{M}$, the largest of the 26 sporadic finite simple groups, to modular functions. The Moonshine module $V^{\natural}$ is a holomorphic vertex operator algebra (VOA) of central charge 24 constructed by Frenkel, Lepowsky and Meurman (a vertex operator algebra is the mathematically rigorous axiomatization of the chiral algebra of a two-dimensional conformal field theory; holomorphic means that the only irreducible module of $V^{\natural}$ is $V^{\natural}$ itself, and the central charge is the parameter $c$ of its Virasoro algebra of conformal symmetries), and $\operatorname{Aut}(V^{\natural})=\mathbb{M}$. For $g\in\mathbb{M}$ the McKay-Thompson series $T_g(q)=\sum_{n\ge 0}(\operatorname{Tr}_{V^{\natural}_n}g)\,q^{n-1}$ collects the graded traces of $g$ on the grading $V^{\natural}=\bigoplus_{n\ge0}V^{\natural}_n$, so that $T_1(q)=j(q)-744$. The Conway-Norton conjecture states that each $T_g$ is a Hauptmodul, i.e. a weight-zero modular function that generates the function field of a genus-zero modular group; Borcherds proved it in the 1992 paper.

His proof ran through Lie theory. A generalized Kac-Moody algebra (also called a Borcherds algebra or Borcherds-Kac-Moody algebra) is a Lie algebra defined by generators and relations encoded in a generalized Cartan matrix whose simple roots may be imaginary (non-positive squared norm) and may occur with multiplicity; like a Kac-Moody algebra it has a Weyl group (the reflection group of its real root system) and a denominator identity, an infinite product-sum identity that is the graded character of the algebra. The monster Lie algebra $\mathfrak{m}$ is obtained from the tensor product $V^{\natural}\otimes V_{\mathrm{II}_{1,1}}$ of the Moonshine module with the lattice VOA of the even unimodular Lorentzian lattice $\mathrm{II}_{1,1}$ (the unique even unimodular lattice of signature $(1,1)$): its space of physical states, identified by the no-ghost theorem from bosonic string theory (the theorem asserting that the physical states of the 26-dimensional bosonic string form a positive semi-definite space), forms a $\mathbb{Z}^2$-graded Borcherds algebra of rank 2 with root lattice $\mathrm{II}_{1,1}$, whose denominator identity is the product expansion of the modular invariant $j$-function, and whose graded traces realize every McKay-Thompson series.

Replacing $V^{\natural}$ by the Leech lattice VOA $V_{\Lambda}$ (where $\Lambda$ is the unique even unimodular positive-definite lattice of rank 24 with no vectors of norm 2, and $\operatorname{Aut}(\Lambda)=\mathit{Co}_0\cong 2\cdot\mathit{Co}_1$ is the Conway group) yields the fake monster Lie algebra, of rank 26 and root lattice $\mathrm{II}_{25,1}$. Twisting the fake monster denominator identity by the standard lift to $\operatorname{Aut}(\hat{\Lambda})=2^{24}.2.\mathit{Co}_1$ (a nonsplit extension $2^{24}.\operatorname{Aut}(\Lambda)$, where $\hat{\Lambda}$ is the standard double cover of $\Lambda$) of an element $\nu\in\operatorname{Aut}(\Lambda)$ produces a further family of Borcherds-Kac-Moody (super)algebras (denoted $\mathfrak{g}_{\phi_\nu}$ in the later literature); for a prime $p\in\{2,3,5,7,11,23\}$ with $(p+1)\mid 24$, taking $\nu$ of cycle shape $1^{24/(p+1)}p^{24/(p+1)}$ gives algebras which Borcherds observed "seem to correspond", in the same loose sense in which the fake monster corresponds to the Monster, to sporadic groups: the baby monster for $p=2$, the Fischer group $\mathit{Fi}_{24}'$ for $p=3$, the Harada-Norton group for $p=5$, the Held group for $p=7$, and $M_{12}$ for $p=11$. The $p=2$ algebra, of rank 18 and root lattice $\Lambda_2\oplus\mathrm{II}_{1,1}$ where $\Lambda_2$ is the 16-dimensional Barnes-Wall lattice, is the baby monster Lie algebra of the questions below (now usually called the fake baby monster Lie algebra). In modern terminology (Carnahan), for every $g\in\mathbb{M}$ the $g$-twisted module $V^{\natural}_g$ yields a monstrous Lie algebra $\mathfrak{m}_g$; $g$ is called Fricke if $T_g$ is proportional to its image under the involution $\tau\mapsto -1/(N\tau)$ for some positive integer $N$ (equivalently, $T_g$ has a pole at $\tau=0$).

A natural construction means a realization of such an algebra other than by generators and relations: typically as the BRST or semi-infinite cohomology of a vertex algebra of central charge 26, which is the algebraic operation by which bosonic string theory quantizes a classical solution space into the Lie algebra of its physical states. The finite group attached to the construction should act on the resulting algebra; for the baby monster Lie algebra this is the double cover $2\cdot B$ of the baby monster finite simple group $B$ (the second-largest sporadic group).

A modular function $f$ is replicable if its images under replication, a modification of Hecke operators, satisfy compatible recursions, and completely replicable if all its replicates are again replicable; every McKay-Thompson series is completely replicable. Norton conjectured that every nontrivial completely replicable function is a Hauptmodul. Finally, $\mathrm{II}_{9,1}$ and $\mathrm{II}_{17,1}$ denote the even unimodular Lorentzian lattices of ranks 10 and 18, isomorphic to $E_8\oplus\mathrm{II}_{1,1}$ and $(E_8\oplus E_8)\oplus\mathrm{II}_{1,1}$; the lattice VOAs of $E_8$ and $E_8\oplus E_8$ are the two obvious non-Leech inputs to the same physical-state construction, and the real simple roots of the resulting Lie algebras are the Dynkin diagrams of the reflection groups of these lattices, which were described by Vinberg.

Problem Statement

This problem is the open remainder of the eight questions Borcherds posed in Section 15 of "Monstrous moonshine and monstrous Lie superalgebras" (1992) about the monster Lie algebra $\mathfrak{m}$, the fake monster Lie algebra, their twisted variants, and their modular identities. The following parts of the original list are already established and may be used as known results: question (1), the modular-form assumptions of Section 14 Example 2 for the primes $p=2,3,5,7,11,23$; question (4), the explicit vertex-algebraic description of the Lie bracket of $\mathfrak{m}$; and question (7) in its non-degenerate case, the genus-zero property of non-degenerate completely replicable functions, together with their complete enumeration (328 functions, three of them the modular fictions $q^{-1}$ and $q^{-1}\pm q$).

The open core to be resolved is the following, retaining the original generality.

(i) Natural constructions: give, for the twisted fake-monster Lie (super)algebras beyond the cases already realized (the ten squarefree-level cases of Möller's BRST construction, which include the fake monster itself and the fake baby monster, together with Scheithauer's rank-10 fake monster superalgebra; in these cases the Lie (super)algebra itself is realized, but the expected finite-group action is not yet exhibited in general), a construction other than by generators and relations, and show that the expected finite groups act on the resulting algebras; in particular, exhibit an action of the double cover $2\cdot B$ of the baby monster on the baby monster Lie algebra.

(ii) Embeddings: determine whether the baby monster Lie algebra (the $p=2$ algebra, of rank 18, with root lattice $\Lambda_2\oplus\mathrm{II}_{1,1}$ for the 16-dimensional Barnes-Wall lattice $\Lambda_2$) is a subalgebra of the monster Lie algebra $\mathfrak{m}$ in a way that preserves the action of $2\cdot B$; and likewise whether the other monstrous Lie algebras (in the modern sense, the Carnahan algebras $\mathfrak{m}_g$ for $g\in\mathbb{M}$) can be realized as subalgebras of $\mathfrak{m}$.

(iii) Denominator identities: investigate the Lie algebras and superalgebras coming from the other elements of the Monster or of $\operatorname{Aut}(\hat{\Lambda})$, covering both Fricke and non-Fricke elements, and write down their denominator formulas explicitly in closed infinite-product form (a product over the roots with the root multiplicities given explicitly, of the same type as the fake monster denominator identity).

(iv) Explicit generalized Kac-Moody algebras: find generalized Kac-Moody algebras outside the finite-dimensional, affine, monstrous and fake-monstrous families for which both the simple roots and the root multiplicities can be described explicitly. Affirmative answers are known (Niemann's six algebras with all root multiplicities explicit, and the gnome Lie algebra); the search for further such algebras beyond these families remains open.

(v) The $E_8$ and $E_8\oplus E_8$ algebras: describe the Lie algebras arising from the vertex algebras of the $E_8$ and $E_8\oplus E_8$ lattices, whose root lattices are $\mathrm{II}_{9,1}$ and $\mathrm{II}_{17,1}$: determine the full set of imaginary simple roots (the naive level-1 pattern is known to fail from norm $-10$ in the $\mathrm{II}_{9,1}$ case) and give closed-form root multiplicities.

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