Generalization of Calabi-Yau/Landau-Ginzburg correspondence

We discuss a possible generalization of the Calabi-Yau/Landau-Ginzburgcorrespondence to a more general class of manifolds. Specifically we considerthe Fermat type hypersurfaces $M_N^k$: $\sum_{i=1}^N X_i^k =0$ in ${\bfCP}^{N-1}$ for various values of k and N. When k2. We assumethat this massless sector is described by a Landau-Ginzburg (LG) theory ofcentral charge $c=3N(1-2/k)$ with N chiral fields with U(1) charge $1/k$. Wecompute the topological invariants (elliptic genera) using LG theory andmassive vacua and compare them with the geometrical data. We find that theresults agree if and only if k=even and N=even. These are the cases when the hypersurfaces have a spin structure. Thus wefind an evidence for the geometry/LG correspondence in the case of spinmanifolds.Comment: 19 pages, Late