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

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 $CP^{N-1}$.