Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space

TL;DR

This work extends mirror symmetry to Calabi–Yau complete intersections in weighted projective spaces with a single Kähler modulus by constructing explicit mirrors via orbifold quotients and desingularization, and by deriving the associated Picard–Fuchs equations for period computations. It then extracts the mirror map and Yukawa couplings from period data, producing explicit instanton numbers $n_d$ for two concrete one-modulus models and detailing the topological data of the mirrors (e.g., $h_{1,1}$ and $\chi$). For three additional models without explicit mirrors, the authors extrapolate parameter data to predict integer $n_d$ values, demonstrating the robustness of the method and providing practical enumerative invariants for these weighted-projective complete intersections. Overall, the paper broadens the computational reach of mirror symmetry to a rich class of Calabi–Yau geometries, enabling precise calculations of worldsheet instanton corrections and related enumerative invariants via the mirror correspondence.

Abstract

We consider a class of Calabi-Yau compactifications which are constructed as a complete intersection in weighted projective space. For manifolds with one Kähler modulus we construct the mirror manifolds and calculate the instanton sum.