On the Structure of the Small Quantum Cohomology Rings of Projective Hypersurfaces

Abstract

We give an explicit procedure which computes for degree $d\le 3$ the correlation functions of topological sigma model (A-model) on a projective Fano hypersurface $X$ as homogeneous polynomials of degree $d$ in the correlation functions of degree 1 (number of lines). We extend this formalism to the case of Calabi-Yau hypersurfaces and explain how the polynomial property is preserved. Our key tool is the construction of universal recursive formulas which express the structural constants of the quantum cohomology ring of $X$ as weighted homogeneous polynomial functions in the constants of the Fano hypersurface with the same degree and dimension one more. We propose some conjectures about the existence and the form of the recursive formulas for the structural constants of rational curves of arbitrary degree. Our recursive formulas should yield the coefficients of the hypergeometric series used in the mirror calculation. Assuming the validity of the conjectures we find the recursive laws for rational curves of degree 4 and 5.

32 pages, changed fonts, exact results on quintic rational curves are added. To appear in Commun. Math. Phys

References in corpus (8)

• Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties
• Mirror Manifolds in Higher Dimension
• Equivariant Gromov - Witten Invariants
• Quantum cohomology of complete intersections
• The quantum cohomology of homogeneous varieties
• Mirror Symmetry and An Exact Calculation of N-2 Point Correlation Function on Calabi-Yau Manifold embedded in CP^{N-1}
• On Quantum Cohomology Rings for Hypersurfaces in $C{P}^{N-1}$
• Construction of Free Energy of Calabi-Yau manifold embedded in $C{P}^{n-1}$ via Torus Actions

Cited by in corpus (19)

• Quantum K theory of symplectic Grassmannians
• Gromov-Witten invariants of Fano threefolds of genera 6 and 8
• Gauss-Manin System and the Virtual Structure Constants
• Symplectic fillings of asymptotically dynamically convex manifolds II--$k$-dilations
• GLSMs for exotic Grassmannians
• Completion of the Conjecture: Quantum Cohomology of Fano Hypersurfaces
• Big quantum cohomology of Fano complete intersections
• Genus one GW invariants of quintic threefolds via MSP localization
• Virtual Structure Constants as Intersection Numbers of Moduli Space of Polynomial Maps with Two Marked Points
• Open Virtual Structure Constants and Mirror Computation of Open Gromov-Witten Invariants of Projective Hypersurfaces
• Geometrical Proof of Generalized Mirror Transformation of Projective Hypersurfaces
• Virtual Gromov-Witten Invariants and the Quantum Cohomology Rings of General Type Projective Hypersurfaces
• Direct Proof of Mirror Theorem of Projective Hypersurfaces up to degree 3 Rational Curves
• Hori's Equation for Gravitational Virtual Structure Constants of Calabi-Yau Hypersurface in CP^{N-1}
• Quantum cohomology of smooth complete intersections in weighted projective spaces and singular toric varieties
• Quantum cohomology from mixed Higgs-Coulomb branches
• Elliptic Virtual Structure Constants and Generalizations of BCOV-Zinger Formula to Projective Fano Hypersurfaces
• Period Integrals (Givental's I-function) of Calabi-Yau Hypersurface in $C{P}^{N-1}$ and Intersection Numbers of Moduli Space of Quasimaps from $C{P}^1$ with Two Marked Points to $C{P}^{N-1}$
• Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation