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

Abstract

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

32 pages, Revised version

References in corpus (10)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
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• Mirror Manifolds in Higher Dimension
• Equivariant Gromov - Witten Invariants
• The Topological CP^1 Model and the Large-N Matrix Integral
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• The quantum cohomology of homogeneous varieties
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Cited by in corpus (12)

• Equivariant Gromov - Witten Invariants
• On the Fukaya category of a Fano hypersurface in projective space
• On the Structure of the Small Quantum Cohomology Rings of Projective Hypersurfaces
• On the work of Givental relative to mirror symmetry
• Gauss-Manin System and the Virtual Structure Constants
• On the Quantum Cohomology Rings of General Type Projective Hypersurfaces and Generalized Mirror Transformation
• GLSMs for exotic Grassmannians
• Completion of the Conjecture: Quantum Cohomology of Fano Hypersurfaces
• Geometrical Proof of Generalized Mirror Transformation of Projective Hypersurfaces
• Disk Amplitudes in Topological A-Model on Calabi-Yau Spaces and Fano Manifolds
• Virtual Gromov-Witten Invariants and the Quantum Cohomology Rings of General Type Projective Hypersurfaces
• On the semi-simplicity of the quantum cohomology algebras of complete intersections