Mirror symmetry and automorphisms

Abstract

We show that there is an extra grading in the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-HÜbsch. Their duality matches cohomology classes of two Calabi-Yau orbifolds. When both orbifolds are equipped with an automorphism s of the same order, our mirror duality involves the weight of the action of on cohomology. In particular it matches the respective s-fixed loci, which are not Calabi-Yau in general. When applied to K3 surfaces with nonsymplectic automorphism s of odd prime order, this provides a proof that Berglund-HÜbsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.