[Paper Review] Mirror Symmetry

Eric Zaslow, Ravi Vakil+6 more|arXiv (Cornell University)|Jan 1, 2003

Black Holes and Theoretical PhysicsPhysics and Astronomy1,029 citations

TL;DR

This paper proves mirror symmetry for 1+1 dimensional supersymmetric sigma models on Kähler manifolds by showing the equivalence between a gauged linear sigma model with enlarged gauge symmetry and a Toda-type Landau-Ginzburg theory. Key mechanisms include R → 1/R duality and vortex-induced superpotential generation, establishing mirror symmetry for both Calabi-Yau and positive first Chern class manifolds, including those deformed by holomorphic isometries.

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.

Motivation & Objective

• To establish mirror symmetry for supersymmetric sigma models on Kähler manifolds in 1+1 dimensions.
• To extend mirror symmetry beyond Calabi-Yau manifolds to include those with positive first Chern class.
• To incorporate deformations of the action by holomorphic isometries into the mirror symmetry framework.
• To demonstrate the equivalence between a gauged linear sigma model with enlarged gauge symmetry and a Toda-type Landau-Ginzburg theory.

Proposed method

• Utilizes a gauged linear sigma model embedded in a theory with enlarged gauge symmetry to facilitate duality analysis.
• Applies standard R → 1/R duality to relate different regimes of the gauge theory.
• Employs dynamical generation of the superpotential via vortex condensation in the low-energy effective theory.
• Establishes equivalence between the gauged linear sigma model and a Toda-type Landau-Ginzburg model through duality and symmetry analysis.
• Analyzes the resulting superpotential and vacuum structure to confirm mirror symmetry correspondence.
• Considers both local and global Calabi-Yau manifolds as well as manifolds with positive first Chern class, including deformed actions via holomorphic isometries.

Experimental results

Research questions

• RQ1 Can mirror symmetry be rigorously proven for 1+1 dimensional supersymmetric sigma models on general Kähler manifolds?
• RQ2 How does the inclusion of positive first Chern class manifolds affect the mirror symmetry correspondence?
• RQ3 What role do holomorphic isometries play in deforming the action while preserving mirror symmetry?
• RQ4 How do vortex condensation and R → 1/R duality contribute to the emergence of the superpotential in the dual theory?
• RQ5 Is the equivalence between the gauged linear sigma model and a Toda-type Landau-Ginzburg theory sufficient to establish mirror symmetry in this context?

Key findings

• Mirror symmetry is rigorously proven for supersymmetric sigma models on Kähler manifolds in 1+1 dimensions.
• The equivalence between the gauged linear sigma model with enlarged gauge symmetry and a Toda-type Landau-Ginzburg theory is established as the core duality mechanism.
• The R → 1/R duality plays a central role in connecting dual phases of the theory.
• Dynamical superpotential generation via vortices is essential for realizing the mirror map in the low-energy effective theory.
• The framework successfully extends mirror symmetry to manifolds with positive first Chern class, including those with holomorphic isometry deformations.
• The results confirm mirror symmetry for both local and global Calabi-Yau manifolds as special cases of the broader construction.