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Fano-Mathieu correspondence
Cornell University
,
2018.
No. 1809.02738.
Galkin S.
We show that G-Fano threefolds are mirror-modular.
1. Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of SL2(ℝ).
2. Quantum periods, shifted by an integer constant (eigenvalue of quantum operator on primitive cohomology) are expansions of weight 2 modular forms (theta-functions) in terms of inversed Hauptmoduln.
3. Products of inversed Hauptmoduln with some fractional powers of shifted quantum periods are very nice cuspforms (eta-quotients).
The latter cuspforms also appear in work of Mason and others: they are eta-products, related to conjugacy classes of sporadic simple groups, such as Mathieu group M24 and Conway's group of isometries of Leech lattice.
This gives a strange correspondence between deformation classes of G-Fano threefolds and conjugacy classes of Mathieu group M24.
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