Fano-Mathieu correspondence

We show that $G$ -Fano threefolds are mirror-modular.

Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of $S L_{2} (R)$ .
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.
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 $M_{24}$ 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 $M_{24}$ .

We show that $G$ -Fano threefolds are mirror-modular.

Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of $S L_{2} (R)$ .
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.
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 $M_{24}$ 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 $M_{24}$ .

文献摘要