The paper by Hori and Vafa provides a proof of mirror symmetry for supersymmetric sigma models on Kähler manifolds in 1+1 dimensions. The proof involves establishing the equivalence between the gauged linear sigma model and a Landau-Ginzburg theory of Toda type, with standard $R \rightarrow 1/R$ duality and dynamical generation of the superpotential by vortices playing crucial roles. This not only confirms mirror symmetry for (local and global) Calabi-Yau manifolds but also extends it to sigma models on manifolds with positive first Chern class, including deformations by holomorphic isometries. The authors use an analogous idea to study the long-distance behavior of (2,2) gauge theories, dualizing the phase of charged fields and describing the low-energy effective theory in terms of dual variables. They demonstrate that a superpotential is dynamically generated by instanton effects, leading to an exact determination of the (twisted) F-term part of the effective Lagrangian. The paper also discusses the mirror symmetry of non-compact Calabi-Yau manifolds and complete intersections in toric varieties, providing a comprehensive framework for understanding mirror symmetry in various contexts.The paper by Hori and Vafa provides a proof of mirror symmetry for supersymmetric sigma models on Kähler manifolds in 1+1 dimensions. The proof involves establishing the equivalence between the gauged linear sigma model and a Landau-Ginzburg theory of Toda type, with standard $R \rightarrow 1/R$ duality and dynamical generation of the superpotential by vortices playing crucial roles. This not only confirms mirror symmetry for (local and global) Calabi-Yau manifolds but also extends it to sigma models on manifolds with positive first Chern class, including deformations by holomorphic isometries. The authors use an analogous idea to study the long-distance behavior of (2,2) gauge theories, dualizing the phase of charged fields and describing the low-energy effective theory in terms of dual variables. They demonstrate that a superpotential is dynamically generated by instanton effects, leading to an exact determination of the (twisted) F-term part of the effective Lagrangian. The paper also discusses the mirror symmetry of non-compact Calabi-Yau manifolds and complete intersections in toric varieties, providing a comprehensive framework for understanding mirror symmetry in various contexts.