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Proof of the umbral moonshine conjecture. (English) Zbl 1383.11052

Summary: The umbral moonshine conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. T. Gannon [Bull. Lond. Math. Soc. 38, No. 1, 1–33 (2006; Zbl 1100.11016)] has proved this for the special case involving the largest sporadic simple Mathieu group. Here, we establish the existence of the umbral moonshine modules in the remaining 22 cases.

MSC:

11F22 Relationship to Lie algebras and finite simple groups
11F37 Forms of half-integer weight; nonholomorphic modular forms
17B69 Vertex operators; vertex operator algebras and related structures

Citations:

Zbl 1100.11016

Software:

SageMath
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References:

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