Abstract. The Landau-Ginzburg Mirror Symmetry Conjecture states that for a quasi-homogeneous singularity W and a group G of symmetries of W , there is a dual singularity W T such that the orbifold A-model of W/G is isomorphic to the B-model of W T . The Landau-Ginzburg A-model is the Frobenius algebra H W,G constructed by Fan, Jarvis, and Ruan, and the B-model is the orbifold Milnor ring of W T . We verify the Landau-Ginzburg Mirror Symmetry Conjecture for Arnol'd's list of unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group, and include a discussion of eight axioms which facilitate the computation of FJRW-rings.

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...

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In this paper we consider the orbifold curve, which is a quotient of an elliptic curve E by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov-Witten theory of the orbifold curve via the product of the Gromov-Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Giventals action giving the CY/LG correspondence between the Gromov-Witten theory of the orbifold curve E /Z 4 and FJRW theory of the pair defined by the polynomial x 4 + y 4 + z 2 and the maximal group of diagonal symmetries. The methods we have developed can easily be applied to other finite quotients of an elliptic curve. Using Givental's action we also recover this FJRW theory via the product of the Gromov-Witten theories of a point. Combined with the CY/LG action we get a result in pure Gromov-Witten theory with the help of modern mirror symmetry conjectures. 4,4,2 39References 40 ALEXEY BASALAEV AND NATHAN PRIDDIS pairing η. A cohomological field theory (CohFT for brevity) Λ g,n on (V, η) is a system of linear maps Λ g,n

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