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On Berenstein-Douglas-Seiberg Duality

Volker Braun

TL;DR

This work formalizes Seiberg duality through the Berenstein–Douglas criterion: two quivers are dual precisely when their bounded derived categories of representations are equivalent, $D^b(Q_1\text{-mod}) \simeq D^b(Q_2\text{-mod})$. It provides concrete demonstrations using both a toy quiver and physically motivated quivers with superpotential, showing that the duality is encoded via tilting complexes and, in favorable cases, Fourier–Mukai transforms. The author shows that toric duals are compatible with BDS duality by constructing explicit tilting complexes whose endomorphism algebras reproduce the dual quivers and F-term constraints. In addition, invariants such as global dimension and K‑theory are discussed as obstructions to duality, illustrating how derived-category methods distinguish dual from non-dual pairs. Overall, the paper places Seiberg duality in a rigorous derived-category framework with practical methods for verifying dualities in quiver gauge theories.

Abstract

I review the proposal of Berenstein-Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the requirements. Then I explicitly show that a pair of toric dual quivers is also dual according to their proposal. All these computations go beyond tilting modules, and really work in the derived category. I introduce all necessary mathematics where needed.

On Berenstein-Douglas-Seiberg Duality

TL;DR

This work formalizes Seiberg duality through the Berenstein–Douglas criterion: two quivers are dual precisely when their bounded derived categories of representations are equivalent, . It provides concrete demonstrations using both a toy quiver and physically motivated quivers with superpotential, showing that the duality is encoded via tilting complexes and, in favorable cases, Fourier–Mukai transforms. The author shows that toric duals are compatible with BDS duality by constructing explicit tilting complexes whose endomorphism algebras reproduce the dual quivers and F-term constraints. In addition, invariants such as global dimension and K‑theory are discussed as obstructions to duality, illustrating how derived-category methods distinguish dual from non-dual pairs. Overall, the paper places Seiberg duality in a rigorous derived-category framework with practical methods for verifying dualities in quiver gauge theories.

Abstract

I review the proposal of Berenstein-Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the requirements. Then I explicitly show that a pair of toric dual quivers is also dual according to their proposal. All these computations go beyond tilting modules, and really work in the derived category. I introduce all necessary mathematics where needed.

Paper Structure

This paper contains 19 sections, 7 theorems, 28 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Review of the BDS--Duality Conjecture
  3. Beyond Representations
  4. The Conjecture
  5. The Berenstein--Douglas Toy Quiver
  6. Path Algebra, Representations and Modules
  7. The Derived Category
  8. Computations in the Derived Category
  9. Fourier--Mukai Transformation
  10. Physical Quivers
  11. Quivers with Superpotential
  12. A Sample Duality Pair
  13. Beyond Tilting Modules
  14. Toric Duality is Seiberg Duality
  15. Review of Toric Duality
  16. ...and 4 more sections

Key Result

Theorem 1

Let $Q$ be an arbitrary quiver. Then the category of representations is the same as the category of ${\mathbb{C}^{}}Q$-modules:

Figures (1)

  • Figure 1: $X_1$ and $X_2$ are subvarieties of $Z_1$ and $Z_2$, which are both resolutions of $Z$

Theorems & Definitions (17)

  • Conjecture 1: Berenstein--Douglas
  • Definition 1
  • Example 1
  • Theorem 1
  • Example 2
  • Theorem 2
  • Theorem 3
  • Definition 2
  • Definition 3
  • Theorem 4
  • ...and 7 more