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Umbral moonshine conjecture for the modules KXK^X

For each of the 23 umbral cases, let GXG^X be the corresponding finite group, let mm be its associated positive integer, and let IX⊂{1,2,…,m−1}I^X\subset\{1,2,\dots,m-1\} be the specified subset. Consider

KX=⨁r∈IX⨁D≤0\D≡r2(mod4m)Kr,DX.K^X=\bigoplus_{r\in I^X}\bigoplus_{\substack{D\leq0\D\equiv r^2\pmod{4m}}}K^X_{r,D}.

For g∈GXg\in G^X and r∈IXr\in I^X, the component Hg,rXH^X_{g,r} is the graded character corrected by the polar term −2q−1/(4m)-2q^{-1/(4m)} when r=1r=1. Umbral moonshine conjecture. There exists a naturally defined bi-graded, infinite-dimensional GXG^X-module KXK^X such that

Hg,rX=−2q−14mδr,1+∑D≤0\D≡r2(mod4m)∞q−D/(4m)Tr⁡Kr,DX(g).H^X_{g,r}=-2q^{-\frac{1}{4m}}\delta_{r,1}+\sum_{\substack{D\leq0\D\equiv r^2\pmod{4m}}}^{\infty}q^{-D/(4m)}\operatorname{Tr}_{K^X_{r,D}}(g).

This is the uniform module-theoretic formulation of the 23 instances of umbral moonshine, relating finite groups to vector-valued mock modular forms; the statement is presented here as the main conjecture of umbral moonshine.

Sources & referencesView supporting material

Primary source

Vassilis Anagiannis and Miranda C. N. Cheng, “TASI Lectures on Moonshine”, arXiv:1807.00723 (2018).

finite-groupsmock-modular-formsumbral-moonshine

Progress summary

Refreshed
Solved

The conjecture was proved in 2015 for all 23 cases, after the Mathieu case had already been settled separately.

The conjecture, formulated in 2012, asserts that the mock modular forms in all 2323 umbral cases arise as graded traces of naturally defined modules.

Known results

  • Gannon proved the M24M_{24} case, corresponding to X=A124X=A_1^{24}.
  • Duncan, Griffin, and Ono proved existence of the modules in the remaining 2222 cases in 2015.
  • The proof identifies the modules through Fourier coefficients, proves nonnegative integral multiplicities, and completes the remaining checks computationally.
  • A 2017 reformulation explicitly confirms existence of all bi-graded modules KXK^X satisfying the conjectured trace identities.

2015 proof and later confirmation

The 2015 paper states that the umbral moonshine modules exist; contemporary and later expositions describe this as a proof of all 2323 moonshines. No retrieved source reports a counterexample, gap, retraction, or subsequent challenge.

Current status (as of August 2026): The existence and required graded-trace identities for all 2323 modules KXK^X are settled; no unresolved mathematical issue was found in the retrieved record.

Sources

Solutions 0

No solutions have been posted yet.