On the Quantum Cohomology Rings of General Type Projective Hypersurfaces and Generalized Mirror Transformation

TL;DR

This paper investigates the quantum cohomology rings of general type projective hypersurfaces, revealing connections with symmetric functions and proposing a generalized mirror transformation to predict Gromov-Witten invariants.

Contribution

It introduces a generalized mirror transformation for hypersurfaces with negative first Chern class and constructs a resolution of the moduli space of polynomial maps.

Findings

Mirror transformation relates quantum cohomology to symmetric functions.

Explicit formula for three-point Gromov-Witten invariants up to cubic curves.

Resolution of moduli space aligns with the generalized mirror transformation.

Abstract

In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with non-positive 1st Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective Calabi-Yau hypersurface has a close relation with the ring of symmetric functions, or with Schur polynomials. With this result in mind, we propose a generalized mirror transformation on the quantum cohomology of a hypersurface with negative first Chern class and construct an explicit prediction formula for three point Gromov-Witten invariants up to cubic rational curves. We also construct a projective space resolution of the moduli space of polynomial maps, which is in a good correspondence with the terms that appear in the generalized mirror transformation.