Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1996-11-08
Commun.Math.Phys. 206 (1999) 157-183
Physics
High Energy Physics
High Energy Physics - Theory
32 pages, changed fonts, exact results on quintic rational curves are added. To appear in Commun. Math. Phys
Scientific paper
10.1007/s002200050701
We give an explicit procedure which computes for degree $d \leq 3$ 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.
Collino Alberto
Jinzenji Masao
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