On the structure of the small quantum cohomology rings of projective hypersurfaces

Alberto Collino, Masao Jinzenji

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)157-183
Number of pages27
JournalCommunications in Mathematical Physics
Volume206
Issue number1
DOIs
Publication statusPublished - 1999
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Fingerprint

Dive into the research topics of 'On the structure of the small quantum cohomology rings of projective hypersurfaces'. Together they form a unique fingerprint.

Cite this