Existence of Generalized Moonshine

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathbb M$ be the Monster finite simple group, and let $C_{\mathbb M}(g)=\{h\in\mathbb M:hg=gh\}$. Put $\mathfrak H=\{\tau\in\mathbb C:\operatorname{Im}\tau>0\}$ and $q=e^{2\pi i\tau}$. A graded projective representation of $C_{\mathbb M}(g)$ is a $\mathbb Q$-graded complex vector space $V(g)=\bigoplus_{r\in\mathbb Q}V(g)_r$, with finite-dimensional graded pieces, on which the centralizer acts by grading-preserving linear maps specified up to nonzero scalar. A lift $\widetilde h$ chooses a representative linear map for $h$. A genus-zero congruence group means a discrete subgroup $\Gamma\le\mathrm{SL}_2(\mathbb R)$ containing a principal congruence subgroup of $\mathrm{SL}_2(\mathbb Z)$, whose compactified modular curve has genus zero. A Hauptmodul is a meromorphic function generating the function field of that modular curve. Let $J(\tau)=j(\tau)-744=q^{-1}+196884q+\cdots$ be the normalized classical modular invariant. Two holomorphic functions are proportional below if one is a nonzero constant multiple of the other.

Statement recorded in the supplied source.

Do there exist, for every $g\in\mathbb M$, a graded projective representation $V(g)$ of $C_{\mathbb M}(g)$, and, for every commuting pair $g,h\in\mathbb M$, a holomorphic function $Z(g,h,\tau)$ on $\mathfrak H$, satisfying all of Norton's conditions?

1. For a suitable lift $\widetilde h$, \[ Z(g,h,\tau)=\sum_{r\in\mathbb Q} \operatorname{Tr}(\widetilde h|V(g)_r)\,q^{r-1}, \] with convergence to the indicated holomorphic function.
2. For every $x\in\mathbb M$, the functions $Z(xgx^{-1},xhx^{-1},\tau)$ and $Z(g,h,\tau)$ are proportional.
3. Each $Z(g,h,\tau)$ is either constant or a Hauptmodul for a genus-zero congruence group.
4. For every $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb Z)$, \[ Z(g^ah^c,g^bh^d,\tau) \quad\text{is proportional to}\quad Z\!\left(g,h,\frac{a\tau+b}{c\tau+d}\right). \]
5. $Z(g,h,\tau)=J(\tau)$ if and only if $g=h=1$.

These conditions state generalized moonshine for all commuting pairs, not merely the untwisted character associated with one Monster element. The cited work proves the historical conjecture. [S2]

Short English statement

Do graded projective centralizer representations and compatible trace functions exist for every commuting pair in the Monster, with conjugation and modular covariance, the constant-or-Hauptmodul property, and the prescribed identity normalization? Carnahan's cited work establishes this historical conjecture.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1194 (ALG-030). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Scott Huai-Lei Carnahan, Generalized Moonshine IV: Monstrous Lie algebras, arXiv:1208.6254, revised 2016, Section 1.1, Norton's conjecture with conditions (1)--(5). https://arxiv.org/abs/1208.6254.