Elliptic genera of landau-ginzburg models over nontrivial spaces

In this paper, we discuss elliptic genera of (2,2) and (0,2) supersymmetric Landau-Ginzburg models over nontrivial spaces, i.e., nonlinear sigma models on nontrivial noncompact manifolds with superpotential, generalizing old computations in Landau-Ginzburg models over (orbifolds of) vector spaces. For Landau-Ginzburg models in the same universality class as nonlinear sigma models, we explicitly check that the elliptic genera of the Landau-Ginzburg models match that of the nonlinear sigma models, via a Thom class computation of a form analogous to that appearing in recent studies of other properties of Landau-Ginzburg models on nontrivial spaces.