On Quantum Cohomology Rings for Hypersurfaces in $CP^{N-1}$

TL;DR

This paper constructs a polynomial representation of the quantum cohomology ring for hypersurfaces in complex projective space, bridging known models and revealing simple relations for certain degrees.

Contribution

It introduces a new torus action method to explicitly construct quantum cohomology rings for hypersurfaces in projective space, generalizing previous models.

Findings

Interpolation between $CP^{N-2}$ and Calabi-Yau hypersurfaces models

Principal relations are simple for degrees $k \,\leq\, N-2$

Relations align with toric compactification of moduli space

Abstract

Using the torus action method, we construct one variable polynomial representation of quantum cohomology ring for degree $k$ hypersurface in $CP^{N-1}$ . The results interpolate the well-known result of $CP^{N-2}$ model and the one of Calabi-Yau hypersuface in $CP^{N-1}$. We find in $k\leq N-2$ case, principal relation of this ring have very simple form compatible with toric compactification of moduli space of holomorphic maps from $CP^{1}$ to $CP^{N-1}$.