Fano hypersurfaces and Calabi-Yau supermanifolds
Abstract
In this paper, we study the geometrical interpretations associated with Sethi's proposed general correspondence between N = 2 Landau-Ginzburg orbifolds with integral c and N = 2 nonlinear sigma models. We focus on the supervarieties associated with c = 3 Gepner models. In the process, we test a conjecture regarding the superdimension of the singular locus of these supervarieties. The supervarieties are defined by a hypersurface W = 0 in a weighted superprojective space and have vanishing super-first Chern class. Here, W is the modified superpotential obtained by adding as necessary to the Gepner superpotential a boson mass term and/or fermion bilinears so that the superdimension of the supervariety is equal to c. When Sethi's proposal calls for adding fermion bilinears, setting the bosonic part of W (denoted by Wbos) equal to zero defines a Fano hypersurface embedded in a weighted projective space. In this case, if the Newton polytope of Wbos admits a nef partition, then the Landau-Ginzburg orbifold can be given a geometrical interpretation as a nonlinear sigma model on a complete intersection Calabi-Yau manifold. The complete intersection Calabi-Yau manifold should be equivalent to the Calabi-Yau supermanifold prescribed by Sethi's proposal.