Fano-Mathieu correspondence

TL;DR

This paper establishes a deep connection between G-Fano threefolds, modular forms, and sporadic simple groups, revealing mirror symmetry and moonshine phenomena involving Mathieu groups and Hauptmoduln.

Contribution

It demonstrates that G-Fano threefolds are mirror-modular and links their quantum periods to modular forms and moonshine subgroups, uncovering a novel correspondence with Mathieu group conjugacy classes.

Findings

Mirror maps are inversed Hauptmoduln for moonshine subgroups.

Quantum periods relate to weight 2 modular forms as expansions in inversed Hauptmoduln.

Cuspforms formed from Hauptmoduln and quantum periods relate to sporadic groups.

Abstract

We show that $G$-Fano threefolds are mirror-modular. 1. Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of $SL_2(\mathbb{R})$. 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 $M_{24}$ 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 $M_{24}$.