On the Structure of the Small Quantum Cohomology Rings of Projective Hypersurfaces |
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Authors: | Alberto Collino Masao Jinzenji |
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Institution: | Dipartimento di Matematica, Universita' di Torino, Via Carlo Alberto 10, 10123 Torino, Italy.?E-mail: collino@dm.unito.it, IT Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan.?E-mail: jin@hep-th.phys.s.u-tokyo.ac.jp, JP
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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.
Received: 29 November 1996 / Accepted: 15 March 1999 |
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