Equivariant mirror symmetry for footballs

TL;DR

This paper proves equivariant mirror symmetry for footballs, extending previous work on projective lines and weighted projective lines, by establishing the equivalence of R-matrices and higher genus graph sums.

Contribution

It establishes equivariant mirror symmetry for footballs, including R-matrix equivalence and higher genus graph sum consistency, generalizing prior results.

Findings

Proved R-matrix equivalence at large radius limit.

Established isomorphism of R-matrices for general radius.

Demonstrated equality of higher genus graph sums for both models.

Abstract

In this paper, we establish equivariant mirror symmetry for footballs $\mathcal{F}(m,r)$. This extends the results by B. Fang, C.C. Liu and Z. Zong, where the projective line was considered [{\it Geometry \& Topology} 24:2049-2092, 2017], and the results by D. Tang of weighted projective lines, on [arXiv:1712.04836]. More precisely, we prove the equivalence of the $R$-matrices for A-model and B-model at large radius limit, and establish isomorphism for $R$-matrices for general radius. We further demonstrate that the graph sum of higher genus cases are the same for both models, hence establish equivariant mirror symmetry for footballs. In last two sections the large radius limit and equivariant limit are considered, resulting a generealized Bouchard-Mari\~{n}o conjecture and Norbury-Scott conjecture respectively.