Motivic L-Function Identities from CFT and Arithmetic Mirror Symmetry

TL;DR

This paper confirms that mirror pairs of Calabi-Yau threefolds in string theory have identical L-functions for their -motives, linking arithmetic mirror symmetry with conformal field theory via motivic isomorphisms.

Contribution

It demonstrates that -motives of mirror pairs are isomorphic, providing a motivic proof of L-function identities predicted by mirror symmetry.

Findings

-motives of mirror pairs are isomorphic.

L-functions of -motives are identical for mirror pairs.

Motivic approach bypasses the need for explicit mirror constructions of zeta functions.

Abstract

Exactly solvable mirror pairs of Calabi-Yau threefolds of hypersurface type exist in the class of Gepner models that include nondiagonal affine invariants. Motivated by the string modular interpretation established previously for models in this class it is natural to ask whether the arithmetic structure of mirror pairs of varieties reflects the fact that as conformal field theories they are isomorphic. Mirror symmetry in particular predicts that the L-functions of the \Omega-motives of such pairs are identical. In the present paper this prediction is confirmed by showing that the \Omega-motives of exactly solvable mirror pairs are isomorphic. This follows as a corollary from a more general result establishing an isomorphism between nondiagonally and diagonally induced motives in this class of varieties. The motivic approach formulated here circumvents the difficulty that no mirror construction of the Hasse-Weil zeta function is known.