N=2 LANDAU-GINZBURG VS. CALABI-YAU σ-MODELS: NON-PERTURBATIVE ASPECTS

Abstract

We discuss some nonperturbative aspects of the correspondence between N=2 Landau-Ginzburg orbifolds and Calabi-Yau σ-models. We suggest that the correct framework is Deligne’s theory of mixed Hodge structures (closely related to catastrophe theory). We derive a general topological formula for the chiral ring OPE coefficients of any Landau-Ginzburg model, including the absolute normalization. This follows from the identification of spectral flow with Grothendieck’s local duality. Wherever the LG model has a CY interpretation, its OPE coefficients are equal to those of the σ-model as given by intersection theory, including normalization. We discuss at length the tricky case of a number of LG fields greater than c/3+2, presenting explicit examples. In passing, we get many results about the geometry of moduli spaces for such conformal theories. We explain the beautiful algebraic geometry connected with a remarkable model pointed out by Vafa, and its relations with moduli space geometry.