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Duality In Two-Dimensional (2,2) Supersymmetric Non-Abelian Gauge Theories

Kentaro Hori

TL;DR

The paper develops a comprehensive framework for understanding low-energy dynamics of two-dimensional N=(2,2) gauge theories with orthogonal, special orthogonal, and symplectic groups. It identifies regular (Coulomb-branch lifted) theories, establishes a network of non-Abelian dualities across gauge groups and matter contents, and validates these dualities through anomaly matching and chiral-ring considerations. The work connects field-theoretic dualities to linear sigma-model realizations of Calabi–Yau geometries and to mathematical equivalences of derived categories, including ramified covers and conifold transitions, thereby linking quantum field theory, string compactifications, and algebraic geometry. The results extend known 3+1D dualities to 1+1D, reveal rich phase structures, and provide concrete dual pairs with explicit mappings of mesons to singlets and of baryons to twist operators, offering new tools for constructing and comparing Calabi–Yau spaces and their D-brane categories. Overall, the study advances the understanding of non-Abelian dynamics in two dimensions and illuminates deep connections between physics and derived-category mathematics.

Abstract

We study the low energy behaviour of N=(2,2) supersymmetric gauge theories in 1+1 dimensions, with orthogonal and symplectic gauge groups and matters in the fundamental representation. We observe supersymmetry breaking in super-Yang-Mills theory and in theories with small numbers of flavors. For larger numbers of flavors, we discover duality between regular theories with different gauge groups and matter contents, where regularity refers to absence of quantum Coulomb branch. The result is applied to study families of superconformal field theories that can be used for superstring compactifications, with corners corresponding to three-dimensional Calabi-Yau manifolds. This work is motivated by recent development in mathematics concerning equivalences of derived categories.

Duality In Two-Dimensional (2,2) Supersymmetric Non-Abelian Gauge Theories

TL;DR

The paper develops a comprehensive framework for understanding low-energy dynamics of two-dimensional N=(2,2) gauge theories with orthogonal, special orthogonal, and symplectic groups. It identifies regular (Coulomb-branch lifted) theories, establishes a network of non-Abelian dualities across gauge groups and matter contents, and validates these dualities through anomaly matching and chiral-ring considerations. The work connects field-theoretic dualities to linear sigma-model realizations of Calabi–Yau geometries and to mathematical equivalences of derived categories, including ramified covers and conifold transitions, thereby linking quantum field theory, string compactifications, and algebraic geometry. The results extend known 3+1D dualities to 1+1D, reveal rich phase structures, and provide concrete dual pairs with explicit mappings of mesons to singlets and of baryons to twist operators, offering new tools for constructing and comparing Calabi–Yau spaces and their D-brane categories. Overall, the study advances the understanding of non-Abelian dynamics in two dimensions and illuminates deep connections between physics and derived-category mathematics.

Abstract

We study the low energy behaviour of N=(2,2) supersymmetric gauge theories in 1+1 dimensions, with orthogonal and symplectic gauge groups and matters in the fundamental representation. We observe supersymmetry breaking in super-Yang-Mills theory and in theories with small numbers of flavors. For larger numbers of flavors, we discover duality between regular theories with different gauge groups and matter contents, where regularity refers to absence of quantum Coulomb branch. The result is applied to study families of superconformal field theories that can be used for superstring compactifications, with corners corresponding to three-dimensional Calabi-Yau manifolds. This work is motivated by recent development in mathematics concerning equivalences of derived categories.

Paper Structure

This paper contains 35 sections, 276 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $O(1)$ Theories
  3. General Remarks On ${\mathbb Z}_2$ Orbifolds
  4. Massive Fields
  5. Corank 1 Degeneration --- Branched Double Cover Of ${\mathbb C}$ Or Its Orbifolds
  6. Corank 2 Degeneration --- Conifold With $r=0$ And $\theta=\pi$
  7. $O(2)$ Theories
  8. Yang-Mills Theory And QCD With Complex Mass --- Supersymmetry Breaking
  9. Twisted Mass --- Defining The ${\mathbb Z}_2$ Orbifold
  10. Massless QCD --- A Dual Description
  11. Corank 1 Degeneration --- Two Or One Massive Vacua
  12. Corank 2 Degeneration --- Ramified Double Cover Of ${\mathbb C}^2$ Or Its Orbifolds
  13. Orthogonal Groups
  14. The Space Of Classical Vacua
  15. Regularity
  16. ...and 20 more sections

Figures (2)

  • Figure 1: The quantum Kähler moduli space for Hosono-Takagi example.
  • Figure 2: A consistent profile of the electric field in the twisted sector