Mirror Symmetry

TL;DR

This paper proves mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions by establishing an equivalence between gauged linear sigma models and Toda-type Landau-Ginzburg theories, using duality and vortex effects.

Contribution

It provides a rigorous proof of mirror symmetry for a broad class of Kahler manifolds, including Calabi-Yau and positively curved manifolds, via a novel gauge theory approach.

Findings

Established equivalence between gauged linear sigma models and Toda Landau-Ginzburg theories.

Proved mirror symmetry for Calabi-Yau and positive first Chern class manifolds.

Utilized R -> 1/R duality and vortex-induced superpotentials in the proof.

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.