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Non-simply-laced Lie algebras via F theory strings

Loriano Bonora, Raffaele Savelli

TL;DR

The paper investigates how non-simply-laced Lie algebras can arise in F-theory by describing symmetry enhancements with string junctions. It develops junction-based realizations for foldings D2n -> Bn, E6 -> F4, and D4 -> G2, and discusses why a Cn realization remains unavailable in the present brane setup. The approach ties algebraic-geometric singularity data to brane configurations and BPS string states, using Jordan strings and orientifold concepts to make the symmetry of Dynkin diagrams manifest. This provides a concrete, physically intuitive method to realize non-simply-laced algebras in F-theory and paves the way for further phenomenological explorations.

Abstract

In order to describe the appearance in F theory of the non--simply--laced Lie algebras, we use the representation of symmetry enhancements by means of string junctions. After an introduction to the techniques used to describe symmetry enhancement, that is algebraic geometry, BPS states analysis and string junctions, we concentrate on the latter. We give an explicit description of the folding of D_{2n} to B_n of the folding of E_6 to F_4 and that of D_4 to G_2 in terms of junctions and Jordan strings. We also discuss the case of C_n, but we are unable in this case to provide a string interpretation.

Non-simply-laced Lie algebras via F theory strings

TL;DR

The paper investigates how non-simply-laced Lie algebras can arise in F-theory by describing symmetry enhancements with string junctions. It develops junction-based realizations for foldings D2n -> Bn, E6 -> F4, and D4 -> G2, and discusses why a Cn realization remains unavailable in the present brane setup. The approach ties algebraic-geometric singularity data to brane configurations and BPS string states, using Jordan strings and orientifold concepts to make the symmetry of Dynkin diagrams manifest. This provides a concrete, physically intuitive method to realize non-simply-laced algebras in F-theory and paves the way for further phenomenological explorations.

Abstract

In order to describe the appearance in F theory of the non--simply--laced Lie algebras, we use the representation of symmetry enhancements by means of string junctions. After an introduction to the techniques used to describe symmetry enhancement, that is algebraic geometry, BPS states analysis and string junctions, we concentrate on the latter. We give an explicit description of the folding of D_{2n} to B_n of the folding of E_6 to F_4 and that of D_4 to G_2 in terms of junctions and Jordan strings. We also discuss the case of C_n, but we are unable in this case to provide a string interpretation.

Paper Structure

This paper contains 13 sections, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A concise review of symmetry enhancement in F theory
  3. Geometric perspective
  4. String perspective
  5. String junctions
  6. Orthogonal Lie algebras
  7. so(2n) algebras
  8. so(2n-1) algebras
  9. $E_6$ and $F_4$
  10. The $E_6$ algebra
  11. The $F_4$ algebra
  12. The ${\bf G_2}$ algebra
  13. Symplectic algebras

Figures (3)

  • Figure 1: The Jordan strings representing the junctions ${\bf a}_2$ and ${\bf a}_3+{\bf a}_4+{\bf a}_5-{\bf b}-{\bf c}_1$
  • Figure 2: The Jordan strings representing the junctions ${\bf c}_1$ and ${\bf a}_3+{\bf a}_4+{\bf a}_5-{\bf b}-{\bf a}_2$
  • Figure 3: The Jordan strings representing the junctions ${\bf a}_1-{\bf a}_2$ and ${\bf a}_3+{\bf a}_4-{\bf b}-{\bf c}$.