# Mirror symmetry for K3 surfaces

## TL;DR

This paper proves that two different constructions of mirror symmetry for certain K3 surfaces, one based on Landau-Ginzburg models and the other on lattice polarization, are equivalent for a specific class of K3 surfaces with particular automorphisms.

## Contribution

It establishes the equivalence of BHK and LPK3 mirror symmetry for K3 surfaces with specific nonsymplectic automorphisms, unifying two major approaches.

## Key findings

- Both mirror symmetry constructions agree for these K3 surfaces.
- The class of K3 surfaces admits automorphisms of order 4, 8, or 12.
- The proof completes the understanding of mirror symmetry in this context.

## Abstract

For certain K3 surfaces, there are two constructions of mirror symmetry that are very different. The first, known as BHK mirror symmetry, comes from the Landau-Ginzburg model for the K3 surface; the other, known as LPK3 mirror symmetry, is based on a lattice polarization of the K3 surface in the sense of Dolgachev's definition. There is a large class of K3 surfaces for which both versions of mirror symmetry apply. In this class we consider the K3 surfaces admitting a certain purely nonsymplectic automorphism of order 4, 8, or 12, and we complete the proof that these two formulations of mirror symmetry agree for this class of K3 surfaces.