Generalization of Calabi-Yau/Landau-Ginzburg correspondence

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 $M_N^k$: $\sum_{i=1}^N X_i^k =0$ in ${\bf CP}^{N-1}$ 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 expect 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.