String modular motives of mirrors of rigid Calabi-Yau varieties

TL;DR

This paper investigates the modular properties of certain higher-dimensional Fano varieties, linking their motives to string theory and mirror symmetry of rigid Calabi-Yau varieties through explicit L-function comparisons.

Contribution

It introduces a class of Fano varieties of special type and demonstrates their motives' modularity, connecting them to string theory and mirror symmetry of rigid Calabi-Yau varieties.

Findings

L-functions of specific Fano varieties match those of their mirror Calabi-Yau counterparts.

Modular forms derived from rational conformal field theory characterize these motives.

The cubic fourfold's modular form relates to an exactly solvable K3 surface.

Abstract

The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the L-function of their $\Omega-$motives. It is shown that the emerging modular forms are string theoretic in origin, derived from the characters of the underlying rational conformal field theory. The definition of the class of Fano varieties of special type is motivated by the goal to find candidates for a geometric realization of the mirrors of rigid Calabi-Yau varieties. We consider explicitly the cubic sevenfold and the quartic fivefold, and show that their motivic L-functions agree with the L-functions of their rigid mirror Calabi-Yau varieties. We also show that the cubic fourfold is string theoretic, with a modular form that is determined by that of an exactly solvable K3 surface.