2012/10/25 by Todor Milanov, Milanov, Todor, Yefeng Shen +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.1210.6862
openalex publication_date 2012/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A simple elliptic singularity of type EN(1,1) (N=6,7,8) can be described in terms of a marginal deformation of an invertible polynomial W. In the papers \citeKS and \citeMR the authors proved a mirror symmetry statement for some particular choices of W and used it to prove quasi-modularity of Gromov-Witten invariants for certain elliptic orbifold ℙ1s. However, the choice of the polynomial W and its marginal deformation ϕμ are not unique. In this paper, we investigate the global mirror symmetry phenomenon for the one-parameter family W+σϕμ. In each case the mirror symmetry is governed by a certain system of hypergeometric equations. We conjecture that the Saito-Givental theory of W+σϕμ at any special limit σ is mirror to either the Gromov-Witten theory of an elliptic orbifold ℙ1 or the Fan-Jarvis-Ruan-Witten theory of an invertible simple elliptic singularity with diagonal symmetries, and the limits are classified by the Milnor number of the singularity and the j-invariant at the special limit. We prove the conjecture when W is a Fermat polynomial. We also prove that the conjecture is true at the Gepner point σ=0 in all other cases.