Geometrical Proof of Generalized Mirror Transformation of Projective Hypersurfaces
Masao Jinzenji
TL;DR
The paper develops a geometric proof of the generalized mirror transformation for genus-0 Gromov-Witten invariants of degree $k$ hypersurfaces in $CP^{N-1}$ by leveraging a two-point quasimap framework. It introduces the compact toric orbifold $\widetilde{Mp}_{0,2}(N,d)$, its stratification by partitions, and a perturbation space to handle excess intersections, culminating in a precise relation between two-point quasimap intersections $w({\cal O}_{h^a}{\cal O}_{h^b})_{0,d}$ and the usual GW invariants $\langle{\cal O}_{h^a}{\cal O}_{h^b}\rangle_{0,d}$ via a partition-sum correction — the generalized mirror transformation for the small quantum cohomology ring. The approach clarifies the geometric meaning of the mirror transformation by interpreting it as removing non-map quasimap contributions, and it recovers known results in Fano and Calabi–Yau cases while enabling reconstruction in general-type settings. This framework provides a more transparent, non-localization derivation of genus-0 mirror symmetry for hypersurfaces and connects quasimap theory with classical GW computations.
Abstract
In this paper, we propose a geometrical proof of the generalized mirror transformation of genus 0 Gromov-Witten invariants of degree k hypersurface in CP^{N-1}.