Fano-Mathieu correspondence

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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.