$p$-adic Berglund-H\"ubsch Duality

Abstract

Berglund-H\"ubsch duality is an example of mirror symmetry between orbifold Landau-Ginzburg models. In this paper we study a D-module-theoretic variant of Borisov's proof of Berglund-H\"ubsch duality. In the $p$-adic case, the D-module approach makes it possible to endow the orbifold chiral rings with the action of a non-trivial Frobenius endomorphism. Our main result is that the Frobenius endomorphism commutes with Berglund-H\"ubsch duality up to an explicit diagonal operator.