Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Seiberg Duality is an Exceptional Mutation

Christopher P. Herzog

TL;DR

This work addresses the non-uniqueness of low-energy gauge theories from D-branes on del Pezzo singularities by recasting Seiberg duality as an admissible mutation of strongly exceptional collections within the derived category. It introduces Ext^{1,2} and strong helices as the organizing principles that guarantee well-behaved dualities and proves two main theorems: (i) Ext^{1,2} implies a well-split, strong-helix structure, and (ii) the Seiberg dual of a collection generating a strong helix also generates a strong helix. The results unify Seiberg duality with mutations and tilting-like equivalences, providing a robust framework that excludes pathological partial dualities and recovers the original gauge-theory duality in a categorical language. The approach yields a path toward classifying the large equivalence class of del Pezzo gauge theories and connects geometric mutations with gauge-theory dualities in a precise, mathematically tractable way.

Abstract

The low energy gauge theory living on D-branes probing a del Pezzo singularity of a non-compact Calabi-Yau manifold is not unique. In fact there is a large equivalence class of such gauge theories related by Seiberg duality. As a step toward characterizing this class, we show that Seiberg duality can be defined consistently as an admissible mutation of a strongly exceptional collection of coherent sheaves.

Seiberg Duality is an Exceptional Mutation

TL;DR

This work addresses the non-uniqueness of low-energy gauge theories from D-branes on del Pezzo singularities by recasting Seiberg duality as an admissible mutation of strongly exceptional collections within the derived category. It introduces Ext^{1,2} and strong helices as the organizing principles that guarantee well-behaved dualities and proves two main theorems: (i) Ext^{1,2} implies a well-split, strong-helix structure, and (ii) the Seiberg dual of a collection generating a strong helix also generates a strong helix. The results unify Seiberg duality with mutations and tilting-like equivalences, providing a robust framework that excludes pathological partial dualities and recovers the original gauge-theory duality in a categorical language. The approach yields a path toward classifying the large equivalence class of del Pezzo gauge theories and connects geometric mutations with gauge-theory dualities in a precise, mathematically tractable way.

Abstract

The low energy gauge theory living on D-branes probing a del Pezzo singularity of a non-compact Calabi-Yau manifold is not unique. In fact there is a large equivalence class of such gauge theories related by Seiberg duality. As a step toward characterizing this class, we show that Seiberg duality can be defined consistently as an admissible mutation of a strongly exceptional collection of coherent sheaves.

Paper Structure

This paper contains 17 sections, 17 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Reading Map
  4. Gauge Theories from B-Branes
  5. Exceptional Sheaves on del Pezzos
  6. Euler Character
  7. Derived Category
  8. Mutations
  9. Braid Group
  10. Helices
  11. Quiver Gauge Theories
  12. Representations of the Quiver
  13. Seiberg Duality
  14. Proof of Theorem \ref{['sewellsplit']}
  15. Proof of Theorem \ref{['setheorem']}
  16. ...and 2 more sections

Key Result

Theorem 3.5

If $(A,B)$ is an exceptional pair in $D^\flat(\mathcal{B})$, then $(L_A^D B, A)$ and $(B, R_B^D A)$ are exceptional pairs in $D^\flat(\mathcal{B})$.

Figures (2)

  • Figure 1: Well split, five node quivers. The blue dot identifies a special polygon around which all arrows travel counterclockwise.
  • Figure 2: A well split six node quiver. The blue dots label polygons around which only nearby arrows travel counterclockwise. The red dot labels a triangle which can disappear if the nodes are moved. Indeed, if we move nodes 1 and 2 farther apart and 4 and 5 closer together, the triangle with a red dot will disappear and a new triangle will take its place around which all arrows travel counterclockwise.

Theorems & Definitions (38)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5: see for example Section 2.3 of Gorodentsev
  • Proposition 3.6: Proposition 1.8 of NK
  • Proposition 3.7: Section 2.3 of Gorodentsev, Assertion 2.3 of Bondal
  • Theorem 3.8: Theorem 4.1 of Bondal
  • Definition 3.9
  • Remark 3.10
  • ...and 28 more