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

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 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. 1

On the Quantum Cohomology Rings of General Type Projective Hypersurfaces and Generalized Mirror Transformation

“…Let us summarize the results of [3]. In [3], we showed that the structure constants L N,k,d m of QH * e (M k N ), (N − k ≥ 2), can be obtained by applying the recursive formulas in Appendix B,…”

Completion of the Conjecture: Quantum Cohomology of Fano Hypersurfaces

“…In [4], we have constructed the recursive formulas that describe the rational structural constant L N,k,d n of the quantum Kähler sub-ring of M k N , which is the degree k hypersurfaces in CP N −1 , in terms of those of M k N +1 when the degree of the rational curves concerned is no more than 5. These recursive formulas determine the structural constants of the quantum cohomology ring of M k N with the aid of the initial conditions given by the following formula in [5]: (jw + (k − j)), (1.1) where N − k must be greater than 1.…”

“…These recursive formulas determine the structural constants of the quantum cohomology ring of M k N with the aid of the initial conditions given by the following formula in [5]: (jw + (k − j)), (1.1) where N − k must be greater than 1. But in [4], our construction of the recursive formula needs some explicit data of the structural constants coming from the higher degree curves, and the determination of the recursive formulas gets extremely harder as the degree of the rational curves grows.…”

Gauss–manin System and the Virtual Structure Constants

“…On the other hand, we conjectured the recursive formulas that evaluate the structure constants of the quantum Kähler sub-ring of M k N in terms of the ones of M k N +1 when N −k ≥ 2 [5], [11]. These recursive formula is strong enough to determine all the structure constants of M k N in this region.…”

“…To clarify the meaning of the virtual structure constants introduced in [5], we had better introduce the B-model deformation parameter x instead of t and consider the following Gauss-Manin system. We can derive the following equality from the above equations:…”