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D-branes on Singularities: New Quivers from Old

David Berenstein, Vishnu Jejjala, Robert G. Leigh

TL;DR

The paper develops a systematic orbifold-of-an-orbifold method to compute D-brane quivers at discrete (non-Abelian) singularities by using exact sequences $0 \rightarrow N \rightarrow G \rightarrow G/N \rightarrow 0$ and constructing the quiver for $M/N$ before encoding the $G/N$ action and discrete torsion. It provides explicit rules for node splitting/merging and arrow connections, and extends to product groups with a detailed treatment of discrete torsion via group cohomology $H^2$ and the interaction term $H^1\otimes H^1$, including Abelian and non-Abelian $G_2$ cases. The framework is illustrated through a variety of examples in ${\mathbb{C}}^2$ and ${\mathbb{C}}^3$ involving ${\widehat{\mathbb{D}}}_{k}$, ${\widehat{\mathbb{E}}}_{6}$, ${\widehat{\mathbb{E}}}_{7}$, and Delta groups, revealing rich torsion structures and dualities. It also highlights that distinct geometric orbifolds can yield identical quivers yet with different superpotentials, underscoring the practical utility of quivers as a robust encoding of low-energy gauge data and their AdS/CFT implications.

Abstract

In this paper we present simplifying techniques which allow one to compute the quiver diagrams for various D-branes at (non-Abelian) orbifold singularities with and without discrete torsion. The main idea behind the construction is to take the orbifold of an orbifold. Many interesting discrete groups fit into an exact sequence $N\to G\to G/N$. As such, the orbifold $M/G$ is easier to compute as $(M/N)/(G/N)$ and we present graphical rules which allow fast computation given the $M/N$ quiver.

D-branes on Singularities: New Quivers from Old

TL;DR

The paper develops a systematic orbifold-of-an-orbifold method to compute D-brane quivers at discrete (non-Abelian) singularities by using exact sequences and constructing the quiver for before encoding the action and discrete torsion. It provides explicit rules for node splitting/merging and arrow connections, and extends to product groups with a detailed treatment of discrete torsion via group cohomology and the interaction term , including Abelian and non-Abelian cases. The framework is illustrated through a variety of examples in and involving , , , and Delta groups, revealing rich torsion structures and dualities. It also highlights that distinct geometric orbifolds can yield identical quivers yet with different superpotentials, underscoring the practical utility of quivers as a robust encoding of low-energy gauge data and their AdS/CFT implications.

Abstract

In this paper we present simplifying techniques which allow one to compute the quiver diagrams for various D-branes at (non-Abelian) orbifold singularities with and without discrete torsion. The main idea behind the construction is to take the orbifold of an orbifold. Many interesting discrete groups fit into an exact sequence . As such, the orbifold is easier to compute as and we present graphical rules which allow fast computation given the quiver.

Paper Structure

This paper contains 20 sections, 77 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The Construction without Discrete Torsion
  3. New Quivers from Old
  4. Example: ${\mathbb{C}}^2/\widehat{\mathbb{D}}_{k}$
  5. Second Example: ${\mathbb{C}}^2/{\mathbb{Z}}_4$
  6. Product Group Orbifolds
  7. Abelian $G_2$
  8. Non-Abelian $G_2$
  9. Examples: Discrete Subgroups of $SU(3)$
  10. ${\mathbb{C}}^3/[(\widehat{\mathbb{D}}_{k} \times {\mathbb{Z}}_{2n})/{\mathbb{Z}}_2]$
  11. The $N=1$ Quiver
  12. $H^2((\widehat{\mathbb{D}}_{k}\times{\mathbb{Z}}_{2n})/{\mathbb{Z}}_2,U(1)) = {\mathbb{Z}}_2$
  13. $H^2((\widehat{\mathbb{D}}_{k}\times{\mathbb{Z}}_{2n})/{\mathbb{Z}}_2,U(1)) = {\mathbb{Z}}_2 \times {\mathbb{Z}}_2$
  14. ${\mathbb{C}}^3/[(\widehat{\mathbb{E}}_{6} \times {\mathbb{Z}}_{2n})/{\mathbb{Z}}_2]$, ${\mathbb{C}}^3/[(\widehat{\mathbb{E}}_{7} \times {\mathbb{Z}}_{2n})/{\mathbb{Z}}_2]$
  15. ${\mathbb{C}}^2/\widehat{\mathbb{E}}_{6}$ Revisited
  16. ...and 5 more sections

Figures (13)

  • Figure 1: Quiver of $\widehat{\mathbb{D}}_{k}$ singularity.
  • Figure 2: Quiver of $\widehat{\mathbb{A}}_{2k-1}$ singularity with a ${\mathbb{Z}}_2$ action.
  • Figure 3: Quiver of $\widehat{\mathbb{E}}_{6}$ singularity.
  • Figure 4: Quiver of $\widehat{\mathbb{E}}_{7}$ singularity.
  • Figure 5: The quiver of the $\widehat{\mathbb{A}}_{3}$ singularity is obtained as a ${\mathbb{Z}}_2$ automorphism of the quiver of the $\widehat{\mathbb{A}}_{2}$ singularity.
  • ...and 8 more figures