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Symmetries and dualities in non-supersymmetric CHL strings

Bernardo Fraiman, Héctor Parra de Freitas

TL;DR

Fraiman and Parra De Freitas analyze four non-supersymmetric CHL-like heterotic string theories with rank reduction $8$ in $8$ dimensions, and develop a lattice-based framework to classify their tree-level moduli spaces, spectra, and duality structures. They construct a unified $1$-loop partition function and apply a genus-1 exploration algorithm to identify all maximal symmetry enhancements, mapping their $D=9$ uplifts to Coxeter diagrams and detailing the associated T-duality and decompactification limits. Three of the $D=9$ theories admit rich Coxeter descriptions, while the fourth exhibits an $ ext{N}=1$ subsector at strong coupling via a type IIB orientifold with an $O7^+$-plane, suggesting stable non-supersymmetric sectors. An orientifold/S-duality analysis links the non-supersymmetric theory to open-string sectors on mutually BPS branes, offering a Bose-Fermi degeneracy interpretation and illuminating stability properties of these backgrounds. Overall, the work provides a comprehensive, diagrammatic, lattice-based map of symmetry and duality in a controlled non-supersymmetric heterotic landscape with practical implications for understanding stability and moduli dynamics in string theory.

Abstract

We chart the classical moduli space of heterotic strings with broken supersymmetry a la Scherk-Schwarz and gauge group rank reduced by 8 in eight dimensions. This space consists of four connected components, each with its own characteristic spectrum and T-duality group. Three of these components uplift to nine dimensions and can be described as Coxeter polyhedra, allowing an exact characterization of their maximal symmetry enhancements and decompactification limits. We determine the maximal enhancements in the eight dimensional theories using lattice based algorithms in the bosonic formulation, and perform an indepth analysis of their massless spectra. Finally we argue that one component has a supersymmetric $\mathcal{N} = 1$ sector described by BPS objects at strong coupling in a non-supersymmetric version of the type IIB string on $T^2/\mathbb{Z}_2$ with one $O7^+$-plane.

Symmetries and dualities in non-supersymmetric CHL strings

TL;DR

Fraiman and Parra De Freitas analyze four non-supersymmetric CHL-like heterotic string theories with rank reduction in dimensions, and develop a lattice-based framework to classify their tree-level moduli spaces, spectra, and duality structures. They construct a unified -loop partition function and apply a genus-1 exploration algorithm to identify all maximal symmetry enhancements, mapping their uplifts to Coxeter diagrams and detailing the associated T-duality and decompactification limits. Three of the theories admit rich Coxeter descriptions, while the fourth exhibits an subsector at strong coupling via a type IIB orientifold with an -plane, suggesting stable non-supersymmetric sectors. An orientifold/S-duality analysis links the non-supersymmetric theory to open-string sectors on mutually BPS branes, offering a Bose-Fermi degeneracy interpretation and illuminating stability properties of these backgrounds. Overall, the work provides a comprehensive, diagrammatic, lattice-based map of symmetry and duality in a controlled non-supersymmetric heterotic landscape with practical implications for understanding stability and moduli dynamics in string theory.

Abstract

We chart the classical moduli space of heterotic strings with broken supersymmetry a la Scherk-Schwarz and gauge group rank reduced by 8 in eight dimensions. This space consists of four connected components, each with its own characteristic spectrum and T-duality group. Three of these components uplift to nine dimensions and can be described as Coxeter polyhedra, allowing an exact characterization of their maximal symmetry enhancements and decompactification limits. We determine the maximal enhancements in the eight dimensional theories using lattice based algorithms in the bosonic formulation, and perform an indepth analysis of their massless spectra. Finally we argue that one component has a supersymmetric sector described by BPS objects at strong coupling in a non-supersymmetric version of the type IIB string on with one -plane.

Paper Structure

This paper contains 39 sections, 68 equations, 2 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Generalities
  3. Tree-level moduli spaces
  4. Local moduli space
  5. Fundamental regions and charge lattices
  6. Partition functions
  7. Conventions
  8. Unified partition function
  9. Massless and tachyonic fields
  10. Vector class
  11. Spinor class
  12. Scalar class
  13. Maximal symmetry enhancements in $D \geq 8$
  14. Charge lattices and gauge symmetries
  15. Exploration algorithm
  16. ...and 24 more sections

Figures (2)

  • Figure 1: Coxeter diagrams representing the reflection symmetries of the even self-dual lattices $\text{II}_{1,17}$ and $\text{II}_{1,9}$ as well as the odd self-dual lattices $\text{I}_{1,17},...,\text{I}_{1,14}$. These correspond respectively to (a) the two 10D supersymmetric heterotic strings on $S^1$, (b) the 9D CHL string and (c) the 10D non-supersymmetric heterotic strings with rank 16 gauge group on $S^1$ and three related subcritical strings. In each case the nodes generate the Weyl subgroup of the T-duality group, with outer automorphisms corresponding to symmetries of the diagrams themselves. All of these diagrams were originally constructed by Vinberg Vinberg:1972.
  • Figure 2: Orientifold S-duals of the supersymmetric CHL string and the non-supersymmetric $\mathcal{B}_{IIb}^{}$ string in $D = 8$. Supersymmetry breaking in this frame corresponds to conjugation of the bottom $Op$-plane RR charges.