# Mirror symmetry and automorphisms

## TL;DR

This paper extends mirror duality for Calabi-Yau orbifolds by incorporating automorphisms, linking fixed loci and cohomology actions, and applies it to K3 surfaces to unify different mirror symmetry frameworks.

## Contribution

It introduces an enhanced mirror duality involving automorphisms, connecting fixed loci and cohomology actions, and proves a relation between Berglund-Hübsch and K3 lattice mirror symmetry.

## Key findings

- Automorphisms influence mirror symmetry via fixed loci and cohomology actions.
- The new duality matches automorphism weights and fixed loci, not just cohomology classes.
- Application to K3 surfaces confirms the equivalence of different mirror symmetry approaches.

## Abstract

We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-H\"ubsch. 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 $s^*$ on cohomology. In particular, it matches the respective $s$-fixed loci, which are not Calabi-Yau in general. When applied to K3 surfaces with non-symplectic automorphism $s$ of odd prime order, this provides a proof that Berglund-H\"ubsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.