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Exploring the landscape of CHL strings on T^d

Anamaria Font, Bernardo Fraiman, Mariana Graña, Carmen A. Núñez, Héctor Parra De Freitas

TL;DR

This work maps the landscape of CHL heterotic string compactifications on ${T^d/\mathbb{Z}}_2$ for $d\le 2$ by exploiting the Mikhailov lattice ${\mathrm{II}}_{(d)}$ and lattice-embedding techniques to locate maximal gauge-group enhancements. It develops and applies a systematic exploration algorithm, generalizes the nine-dimensional CHL construction to $D=10-d$ dimensions, and uses generalized Dynkin diagrams to identify enhancement patterns, including both simply-laced and non-simply-connected groups that arise at Kac-Moody levels 2 and 1. The paper provides a comprehensive catalog of maximal enhancements in 9D and 8D CHL theories, computes overlattices and fundamental groups to determine global gauge structures, and checks anomaly- and duality-consistency, including a robust match with F-theory on K3 with frozen singularities. The results demonstrate the exhaustiveness of the method for $d=1,2$, connect CHL moduli to Narain-like dualities, and offer a foundation for extending the analysis to $d>2$ and to broader CHL components within the string landscape.

Abstract

Compactifications of the heterotic string on special T^d/Z_2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d+8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II_{(d)}, which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d=1 and 2, and give a list of maximally enhanced points where the U(1)^{d+8} enhances to a rank d+8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E_{10}. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings ofof lattices into the dual of II_{(2)}. Our results easily generalize to d > 2.

Exploring the landscape of CHL strings on T^d

TL;DR

This work maps the landscape of CHL heterotic string compactifications on for by exploiting the Mikhailov lattice and lattice-embedding techniques to locate maximal gauge-group enhancements. It develops and applies a systematic exploration algorithm, generalizes the nine-dimensional CHL construction to dimensions, and uses generalized Dynkin diagrams to identify enhancement patterns, including both simply-laced and non-simply-connected groups that arise at Kac-Moody levels 2 and 1. The paper provides a comprehensive catalog of maximal enhancements in 9D and 8D CHL theories, computes overlattices and fundamental groups to determine global gauge structures, and checks anomaly- and duality-consistency, including a robust match with F-theory on K3 with frozen singularities. The results demonstrate the exhaustiveness of the method for , connect CHL moduli to Narain-like dualities, and offer a foundation for extending the analysis to and to broader CHL components within the string landscape.

Abstract

Compactifications of the heterotic string on special T^d/Z_2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d+8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II_{(d)}, which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d=1 and 2, and give a list of maximally enhanced points where the U(1)^{d+8} enhances to a rank d+8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E_{10}. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings ofof lattices into the dual of II_{(2)}. Our results easily generalize to d > 2.

Paper Structure

This paper contains 31 sections, 149 equations, 5 figures, 10 tables.

Table of Contents

  1. Introduction
  2. The nine-dimensional CHL String
  3. Constructing the theory from the heterotic string
  4. Massless vectors
  5. Maximal enhancements from the Generalized Dynkin diagram
  6. T-duality
  7. The CHL string in $D$ dimensions
  8. Extending the nine-dimensional construction
  9. Generalized Dynkin Diagrams
  10. Exploring the moduli space
  11. Exploration algorithm
  12. Example
  13. Matter states and global data
  14. Computing the overlattice
  15. Computing the fundamental group
  16. ...and 16 more sections

Figures (5)

  • Figure 1: Generalized Dynkin Diagram for the lattice $\text{II}_{1,9}$. The coloring of the nodes 0 and C reflects the fact the the associated states have nonzero momentum and/or winding, as opposed to the white nodes.
  • Figure 2: Generalized Dynkin diagram giving enhancements $({\mathrm{G}}_9)_2 + ({\mathrm{A}}_1)_1$. Green (blue) coloring means that the state has nontrivial momentum number and/or winding only along direction 1 (2). The double border of the $\texttt{C}_2$ node indicates that it corresponds to a vector with $Z^2 = 4$.
  • Figure 3: Generalized Dynkin diagram for $d = 2$ theories with enhancements to algebras with ${\mathrm{C}}_n$ factor. It is obtained from the GDD in figure \ref{['edd101a']} by replacing the node $\texttt{C}_2$ with $\texttt{C}_2'$ as shown in \ref{["c2c2'"]}. Yellow coloring means that the state has nontrivial momentum number and/or winding along directions 1 and 2.
  • Figure 4: Generalized Dynkin diagram for $d = 2$ theories, obtained by replacing the node $0$ by $0'$ as shown in \ref{["00'"]}.
  • Figure 5: Scheme of how deleting nodes in the Dynkin diagrams of maximally enhanced groups with $H = \mathbb{Z}_2 \times \mathbb{Z}_2$ lead to gauge groups with lower rank and also with $H = \mathbb{Z}_2 \times \mathbb{Z}_2$.