Mirror Symmetry

TL;DR

The paper establishes mirror symmetry for 2d N=(2,2) supersymmetric sigma models by showing a dual description of GLSMs as Landau-Ginzburg theories of Toda type, with the dual superpotential generated nonperturbatively by vortices. It analyzes the role of R-symmetries, abelian dualities, and localization to derive exact twisted superpotentials and spectra, then extends the framework to toric, compact, and non-compact geometries including hypersurfaces and complete intersections. By connecting D-brane physics, periods, and Picard-Fuchs equations to LG mirrors, it provides a unified mechanism to compute vacua, chiral rings, and prepotentials across Calabi–Yau and Fano-type geometries. The results generalize mirror symmetry beyond CY manifolds, showing LG mirrors for a broad class of toric and non-compact spaces and offering a robust tool for understanding BPS spectra and quantum cohomology through 2d QFT dualities.

Abstract

We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type. Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation. This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the action by holomorphic isometries.