Global mirror symmetry for invertible simple elliptic singularities (Q332176)

At its most basic, mirror symmetry is a local isomorphism called the mirror map between the complex moduli on the \(B\)-model and the Kähler moduli on the \(A\)-model. The complex moduli space parametrizes deformations of complex structure (of the Calabi-Yau manifold under consideration). The Kähler moduli parametrizes the complexified Kähler classes of the mirror.NEWLINENEWLINEThis local isomorphism was originally defined near special points called the large complex structure limits, which can be thought of as having monodromy that is maximally unipotent. Locally, each such degeneration corresponds to the complexified Kähler cone of the mirror to the degenerating family. Conjecturally, the Kähler cones glue together to form the stringy Kähler moduli space, which should be globally isomorphic to the entire complex moduli.NEWLINENEWLINEOn both sides, there are additional special points, such as the Geepner and conifold points. Different correspondences refer to equivalences of invariants defined for various couples of special points and the term \textit{global mirror symmetry} introduced by \textit{A. Chiodo} and \textit{Y. Ruan} [Invent. Math. 182, No. 1, 117--165 (2010; Zbl 1197.14043)] refers to having a unified picture, i.e., a global isomorphism on the entire complex and stringy Kähler moduli spaces.NEWLINENEWLINEThe paper under review is the 4th in a series of papers by various combinations of the authors and Kravitz and Ruan, where global mirror symmetry for simple elliptic singularities is investigated.NEWLINENEWLINEStart with an invertible simple elliptic singularity (ISES) \(W\). Simple elliptic singularity means that \(W=0\) is the germ of a singularity such that the exceptional divisor of its minimal resolution is an elliptic curve. Such a singularity can always be given by a quasi-homogeneous non-degenerate polynomial \(W\). Invertible means that the exponent matrix of \(W\) is invertible. The classification of such ISES is given in Table 1.1 of the paper and they correspond to Dynkin diagrams of type \(E_6\), \(E_7\) and \(E_8\).NEWLINENEWLINETo such a \(W\), consider its miniversal deformation space \(\mathcal{S}\). This is the (local) complex moduli investigated. The authors consider special limit points \(\sigma\in\mathcal{S}\), namely those for which the corresponding deformation of \(W\) no longer has an isolated critical point at \(0\). They explain how to associate to such \(\sigma\) the Saito-Givental ancestor potential \(\mathcal{A}_W^{SG}(\sigma)\).NEWLINENEWLINEThe global mirror symmetry expectation is introduced in Conjecture 1.3, which states that the mirror of special limit points in \(\mathcal{S}\) should be mirror to either of the two following: {NEWLINENEWLINE - The Fan-Jarvis-Ruan-Witten (FJRW) theory of a simple elliptic singularity.NEWLINENEWLINE- The Gromov-Witten (GW) theory of an elliptic orbifold \(\mathbb{P}^1\). NEWLINENEWLINE} The mirror relationship is described as an equality between the corresponding ancestor potentials. The authors prove that Conjecture 1.3 holds for Geepner points (Theorem 1.4) and for Fermat simple elliptic singularities (Theorem 1.5).NEWLINENEWLINEThe paper is overall well-written and includes a good amount of background information. It is instructive to see the full picture of global mirror symmetry unfolding.