2020/11/02 by Alexey Basalaev, Atsushi Takahashi, Basalaev, Alexey +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2011.01033
openalex publication_date 2020/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any triple of positive integers A' = (a1',a2',a3') and c ∈ ℂ^*, cusp polynomial fA' = x1a1'+x2a2'+x3a3'-c-1x1x2x3 is known to be mirror to Geigle-Lenzing orbifold projective line ℙ1a1',a2',a3'. More precisely, with a suitable choice of a primitive form, Frobenius manifold of a cusp polynomial fA', turns out to be isomorphic to the Frobenius manifold of the Gromov-Witten theory of ℙ1a1',a2',a3'. In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any G - a symmetry group of a cusp polynomial fA', we introduce the Frobenius manifold of a pair (fA',G) and show that it is isomorphic to the Frobenius manifold of the Gromov-Witten theory of Geigle-Lenzing weighted projective line ℙ1A,Λ, indexed by another set A and Λ, distinct points on ℂ∖\0,1\. For some special values of A' with the special choice of G it happens that ℙ1A' ≅ ℙ1A,Λ. Combining our mirror symmetry isomorphism for the pair (A,Λ), together with the "usual" one for A', we get certain identities of the coefficients of the Frobenius potentials. We show that these identities are equivalent to the identities between the Jacobi theta constants and Dedekind eta-function.