Generalization of Calabi-Yau/Landau-Ginzburg correspondence

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.