The mirror symmetry of K3 surfaces with non-symplectic automorphisms of prime order

Abstract

In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, that admit a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund-Hübsch-Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface. IntroductionSince its discovery by physicists nearly 30 years ago, mirror symmetry has been the focus of much interest for both physicists and mathematicians. Although mirror symmetry has been "proven" physically, we have much to learn about the phenomenon mathematically. When we speak of mirror symmetry mathematically, there are many different constructions or rules for determining when a Calabi-Yau manifold is "mirror" to another. The constructions are often formulated in terms of families of Calabi-Yau manifolds. A natural question is whether, in a situation where more than one version can apply, they produce the same mirror (or mirror family). In this article, we consider two versions of mirror symmetry for K3 surfaces, and show that in this case the answer is affirmative, as we might expect.The first version of mirror symmetry of interest to us is known as BHK mirror symmetry. This was formulated by Berglund-Hübsch [10], Berglund-Henningson [9] and Krawitz [22] for Landau-Ginzburg models. Using the ideas of the Landau-Ginzburg/Calabi-Yau correspondence, BHK mirror symmetry also produces a version of mirror symmetry for certain Calabi-Yau manifolds (see Section 2).In the BHK construction, one starts with a quasihomogeneous and invertible polynomial W and a group G of symmetries of W satisfying certain conditions (see Section 2.2 for more details). From this data, we obtain the Calabi-Yau (orbifold) defined as the hypersurface Y W,G = {W = 0} /G. Given an LG pair (W, G), BHK mirror symmetry allows to obtain another LG pair (W T , G T ) satisfying the same conditions, and therefore another Calabi-Yau (orbifold) Y W T ,G T . We say that Y W,G and Y W T ,G T form a BHK mirror pair. In our case, we resolve singularities to obtain K3 surfaces X W,G and X W T ,G T , which we call a BHK mirror pair. When no confusion arise, we will denote these mirror K3 surfaces simply by X and X T , respectively.Another form of mirror symmetry for K3 surfaces, which we will call LPK3 mirror symmetry, is described by Dolgachev in [17]. LPK3 mirror symmetry says that the mirror family of a given K3 surface admitting a polarization by a lattice M is the family of K3 surfaces polarized by the mirror lattice M ∨ . We say that the two K3 surfaces are LPK3 mirror when they are lattice polarized and they belong to LPK3 mirror families (see details in Section 2.1).Returning to the question posed earlier, one can ask whether the BHK mirror symmetry and LPK3 mirror symmetry produce the same mirror. A similar question was considered by Belcastro in [8]. She considers a family of K3 surfaces that arise as (th...

Non-geometric Calabi-Yau backgrounds and K3 automorphisms

“…By construction, the BH mirror of a p-cyclic surface has its defining polynomial of the form W T =x p 1 +f (x 2 ,x 3 ,x 4 ), therefore it also admits an order p automorphism σ T p :x → ζ px . The following theorem was proved for p = 2 by Artebani et al [22] and for p ∈ {3, 5, 7, 13} by Comparin et al [23]. 4…”

“…We will call such surfaces p-cyclic, following [23]. They admit the obvious order p non-symplectic automorphism σ p : x 1 → ζ p x 1 .…”

“…As we will review in section 2, at least when p is a prime number, the lattice-polarized mirror symmetry w.r.t. the invariant sublattice associated with the action of σ p , coincides with the Berglund-Hübsch mirror symmetry [22,23]. An important corollary that we will obtain is that the automorphism σ p of the surface on the one-hand and the corresponding automorphism σ T p of the (Berglund-Hübsch) mirror surface on the other hand act on sub-lattices of Γ 4,20 that are orthogonal to each other, denoted respectively T (σ p ) and T (σ T p ).…”

Heterotic/type II duality and non-geometric compactifications

“…Perturbative type IIA vacua preserving N = 2 supersymmetry in four dimensions can be obtained from compactifications on Calabi-Yau three-folds (CY 3 ), or from orbifold or Gepner-point limits of these. In these cases the underlying worldsheet (c,c) = (9,9) conformal field theory (CFT) has an extended (2, 2) superconformal symmetry and both left-and right-moving R-charges are integer-valued. From the worldsheet perspective, four of the eight space-time supercharges come from the left-movers and the other four from the right-movers.…”

“…The two monodromies γ 1 , γ 2 ∈ O(Γ 4,20 ) satisfying the conditions for N = 2 Minkowski vacua are necessarily of finite order, p 1 , p 2 , with γ p 1 1 = γ p 2 2 = 1 for some integers p 1 , p 2 . In [7], constructions were given for prime orders p 1 , p 2 but recent mathematical results [9,10] allow us to extend the constructions of [7] to non-prime integers p 1 , p 2 . Each duality γ satisfying the conditions above has a fixed locus (i.e.…”