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Duality and Weyl Symmetry of 7-brane Configurations

Tamas Hauer, Amer Iqbal, Barton Zwiebach

TL;DR

The paper formalizes the relation between SL$(2,\mathbb{Z})$ dualities of IIB 7-brane backgrounds and the Lie-algebraic data carried by the branes, introducing the duality group ${\cal D}(w)$ as the set of duality transformations that can be compensated by crossing transformations. It shows that Weyl reflections are realized as invariant brane transpositions acting on the junction lattice, while dualities act by automorphisms of the root lattice and, in general, by Dynkin-diagram automorphisms $\Gamma$. The authors compute ${\cal D}(w)$ and the associated homomorphisms to $\Gamma$ for all finite-type brane configurations ${\bf A_N},{\bf D_N},{\bf E_N}$ and their affine extensions, revealing subtle differences between finite and affine cases, including invariant transpositions that act nontrivially on charges. The results unify brane-monodromy and weight-junction data, clarifying how 4d $\mathcal{N}=2$ spectra transform under dualities and how outer automorphisms of Dynkin diagrams arise from brane dynamics. This provides a comprehensive map between 7-brane backgrounds, their gauge-data, and duality-induced automorphisms relevant for D3-probe theories.

Abstract

Extending earlier results on the duality symmetries of three-brane probe theories we define the duality subgroup of SL(2,Z) as the symmetry group of the background 7-branes configurations. We establish that the action of Weyl reflections is implemented on junctions by brane transpositions that amount to exchanging branes that can be connected by open strings. This enables us to characterize duality groups of brane configurations by a map to the symmetry group of the Dynkin diagram. We compute the duality groups and their actions for all localizable 7-brane configurations. Surprisingly, for the case of affine configurations there are brane transpositions leaving them invariant but acting nontrivially on the charges of junctions.

Duality and Weyl Symmetry of 7-brane Configurations

TL;DR

The paper formalizes the relation between SL dualities of IIB 7-brane backgrounds and the Lie-algebraic data carried by the branes, introducing the duality group as the set of duality transformations that can be compensated by crossing transformations. It shows that Weyl reflections are realized as invariant brane transpositions acting on the junction lattice, while dualities act by automorphisms of the root lattice and, in general, by Dynkin-diagram automorphisms . The authors compute and the associated homomorphisms to for all finite-type brane configurations and their affine extensions, revealing subtle differences between finite and affine cases, including invariant transpositions that act nontrivially on charges. The results unify brane-monodromy and weight-junction data, clarifying how 4d spectra transform under dualities and how outer automorphisms of Dynkin diagrams arise from brane dynamics. This provides a comprehensive map between 7-brane backgrounds, their gauge-data, and duality-induced automorphisms relevant for D3-probe theories.

Abstract

Extending earlier results on the duality symmetries of three-brane probe theories we define the duality subgroup of SL(2,Z) as the symmetry group of the background 7-branes configurations. We establish that the action of Weyl reflections is implemented on junctions by brane transpositions that amount to exchanging branes that can be connected by open strings. This enables us to characterize duality groups of brane configurations by a map to the symmetry group of the Dynkin diagram. We compute the duality groups and their actions for all localizable 7-brane configurations. Surprisingly, for the case of affine configurations there are brane transpositions leaving them invariant but acting nontrivially on the charges of junctions.

Paper Structure

This paper contains 15 sections, 57 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Brane configurations, duality and crossing transformations
  3. Definitions and properties
  4. The action on invariant charges
  5. The action on invariant charges
  6. Implementing Lie algebra Weyl transformations
  7. Weyl transformations as invariant crossing transformations
  8. Further examples of invariant crossing transformations
  9. Duality groups and Dynkin graph automorphisms
  10. Duality groups for finite-type configurations
  11. ${\bf A_{N}}$ configuration: ${\bf A^{N+1}}$
  12. ${\bf H_N}$ configurations: ${\bf A^{N+1}C}$
  13. ${\bf D_N}$ configurations: ${\bf A^{N}BC}$
  14. $\bf E_{N}$ configuration: $\bf A^{N-1}BCC$
  15. Duality groups for affine configurations

Figures (4)

  • Figure 1: Weyl transformations correspond to branch cut moves of the 7-brane configuration.
  • Figure 2: Action of $P_1\ldots P_{n-1}$ on a 7-brane configuration with a string junction of no asymptotic charge. While moving the rightmost brane, the $[p,q]$-charges of the branes change but the invariant charges of the junction do not.
  • Figure 3: An open string between two 7-branes can be transformed into an open string between two mutually local 7-branes by rearranging the branes.
  • Figure 11: ${\bf \widehat{E}_1}$ brane configuration and $\widehat{A}_1$ Dynkin diagram.