Mirror symmetry for K3 surfaces

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.

Mirror Symmetry for Nonabelian Landau-Ginzburg Models

“…Example 2.6. The polynomial from Example 2.4 defined by W = x 4 1 + x 4 2 + x 4 3 + x 4 4 with weights ( 1 4 , 1 4 , 1 4 , 1 4 ) is a Fermat polynomial. An example of a chain polynomial is x 3 1 x 2 + x 2 2 x 3 + x 2 3 with weights ( 1 4 , 1 4 , 1 2 ) and an example of a loop polynomial is x 2 1 x 2 + x 2 2 x 3 + x 2 3 x 1 , which has weights ( 1 3 , 1 3 , 1 3 ).…”

“…Finally, let us briefly mention the articles [1], [4], [7], [8], [16] wherein the authors show that in certain cases the Landau-Ginzburg mirror symmetry agrees with more geometric versions of mirror symmetry, such as mirror symmetry for K3 surfaces and Borcea-Voisin mirror symmetry.…”