Abstract:
We give an explicit procedure which computes for degree d≤ 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 structure constants of the quantum cohomology ring of X as weighted homogeneous polynomial functions of 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 laws for the structure 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 four.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 29 November 1996 / Accepted: 15 March 1999
Rights and permissions
About this article
Cite this article
Collino, A., Jinzenji, M. On the Structure of the Small Quantum Cohomology Rings of Projective Hypersurfaces. Comm Math Phys 206, 157–183 (1999). https://doi.org/10.1007/s002200050701
Issue Date:
DOI: https://doi.org/10.1007/s002200050701