This paper explores the phases of N=2 supersymmetric theories in two dimensions, focusing on the relationship between sigma models based on Calabi-Yau hypersurfaces in weighted projective spaces and Landau-Ginzburg models. By analyzing phase transitions in supersymmetric gauge theories, the author finds a natural connection between these models. The construction allows for the recovery of the known correspondence between these types of models and extends it to include new classes of manifolds and models with (0,2) world-sheet supersymmetry. The paper also predicts physical processes involving changes in the topology of space-time. The paper begins by introducing the field theory background, focusing on N=2 supersymmetry in two dimensions. It discusses the construction of chiral and twisted chiral superfields, the gauge field strength, and the Lagrangians for these models. The author then considers the correspondence between Calabi-Yau and Landau-Ginzburg models, showing how the Calabi-Yau manifold can be described by a sigma model with a Kahler metric, while the Landau-Ginzburg model is described by a superpotential. The paper also discusses the generalization of this correspondence to include more complex Calabi-Yau manifolds and models with (0,2) supersymmetry. The author analyzes the low-energy physics of these models for different values of the Fayet-Iliopoulos D term, r. For large positive r, the model reduces to a sigma model with target space being a Calabi-Yau manifold. For small negative r, the model becomes a Landau-Ginzburg theory, with a unique classical vacuum state governed by a superpotential. The paper also discusses the quantum corrections to the sigma model and Landau-Ginzburg orbifold, showing that these corrections do not qualitatively change the nature of the system. The paper concludes by emphasizing that Landau-Ginzburg and Calabi-Yau models are two different phases of the same system, with Landau-Ginzburg being the analytic continuation of Calabi-Yau to negative Kahler class. The author also discusses the implications of these results for the understanding of supersymmetric theories in two dimensions and the role of topological invariants in these models.This paper explores the phases of N=2 supersymmetric theories in two dimensions, focusing on the relationship between sigma models based on Calabi-Yau hypersurfaces in weighted projective spaces and Landau-Ginzburg models. By analyzing phase transitions in supersymmetric gauge theories, the author finds a natural connection between these models. The construction allows for the recovery of the known correspondence between these types of models and extends it to include new classes of manifolds and models with (0,2) world-sheet supersymmetry. The paper also predicts physical processes involving changes in the topology of space-time. The paper begins by introducing the field theory background, focusing on N=2 supersymmetry in two dimensions. It discusses the construction of chiral and twisted chiral superfields, the gauge field strength, and the Lagrangians for these models. The author then considers the correspondence between Calabi-Yau and Landau-Ginzburg models, showing how the Calabi-Yau manifold can be described by a sigma model with a Kahler metric, while the Landau-Ginzburg model is described by a superpotential. The paper also discusses the generalization of this correspondence to include more complex Calabi-Yau manifolds and models with (0,2) supersymmetry. The author analyzes the low-energy physics of these models for different values of the Fayet-Iliopoulos D term, r. For large positive r, the model reduces to a sigma model with target space being a Calabi-Yau manifold. For small negative r, the model becomes a Landau-Ginzburg theory, with a unique classical vacuum state governed by a superpotential. The paper also discusses the quantum corrections to the sigma model and Landau-Ginzburg orbifold, showing that these corrections do not qualitatively change the nature of the system. The paper concludes by emphasizing that Landau-Ginzburg and Calabi-Yau models are two different phases of the same system, with Landau-Ginzburg being the analytic continuation of Calabi-Yau to negative Kahler class. The author also discusses the implications of these results for the understanding of supersymmetric theories in two dimensions and the role of topological invariants in these models.