Fano hypersurfaces and Calabi-Yau supermanifolds

Abstract

In this paper, we study the geometrical interpretations associated with Sethi's proposed general correspondence between = 2 Landau-Ginzburg orbifolds with integral ĉ and = 2 nonlinear sigma models. We focus on the supervarieties associated with ĉ = 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 = 0 in a weighted superprojective space and have vanishing super-first Chern class. Here, 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 ĉ. When Sethi's proposal calls for adding fermion bilinears, setting the bosonic part of (denoted by bos) equal to zero defines a Fano hypersurface embedded in a weighted projective space. In this case, if the Newton polytope of bos 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.