## Abstract

We discuss a possible generalization of the Calabi-Yau/Landau-Ginzburg correspondence to a more general class of manifolds. Specifically we consider the Fermat type hypersurfaces M_{N}^{k}: Σ_{i=1}^{N} X_{i}^{k} = 0 in ℂℙ^{N-1} for various values of k and N. When k < N, the 1-loop beta function of the sigma model on M_{N}^{k} is negative and we except the theory to have a mass gap. However, the quantum cohomology relation σ^{N-1} = const. σ^{k-1} suggests that in addition to the massive vacua there exists a remaining massless sector in the theory if k > 2. We assume that this massless sector is described by a Landau-Ginzburg (LG) theory of central charge c = 3N(1-2/k) with N chiral fields with U(1) charge 1/k. We compute the topological invariants (elliptic genera) using LG theory and massive vacua and compare them with the geometrical data. We find that the results agree if and only if k = even and N = even. These are the cases when the hypersurfaces have a spin structure. Thus we find an evidence for the geometry/LG correspondence in the case of spin manifolds.

Original language | English |
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Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Journal of High Energy Physics |

Volume | 4 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2000 |

Externally published | Yes |

## Keywords

- Conformal Field Models in String Theory
- Differential and Algebraic Geometry
- Sigma Models
- Topological Field Theories

## ASJC Scopus subject areas

- Nuclear and High Energy Physics