Investigating 9d/8d non-supersymmetric branes and theories from supersymmetric heterotic strings

TL;DR

This work analyzes nine- and eight-dimensional heterotic string theories by exploiting outer automorphisms of Narain charge lattices to produce non-supersymmetric theories from supersymmetric ones via asymmetric $bZ_2$ orbifolds. The authors develop a lattice-based framework, including invariant and dual lattices, to determine the untwisted and twisted sectors needed for modular invariance, read off the resulting spectrum, and identify eight disconnected gauge symmetries in $9d$ that host non-SUSY codimension-two branes. They provide explicit folding patterns (e.g., $A_{2n-1} o C_n$) and enumerate massless representations arising in each case, connecting these $9d$ theories to $8d$ examples and to CHL-string structures. A key outcome is the prediction of a set of non-supersymmetric branes tied to the no-global-symmetry conjecture and the appearance of disconnected gauge groups of the form $G timesbZ_2$. The results broaden the heterotic landscape, offer concrete spectra for rank-reduced theories, and point to future duality, folding, and TMF-inspired explorations.

Abstract

We consider heterotic string theories in nine and eight dimensions. We identify the disconnected part of the spacetime gauge group by studying the outer automorphism of the charge lattices. The absence of the global symmetry indicates the existence of non-supersymmetric codimension two branes. Moreover, we provide a list of gauge groups and matter contents of non-supersymmetric rank-reduced heterotic string theories (a branch corresponding to the $E_8$ string on $S^1$) from the orbifolding of the outer automorphism as well as the fermion parity. We also provide examples in eight dimensions.

Figures (3)

• Figure 1: Relation among heterotic string theories in 10d and 9d. Here black arrow corresponds to simple $S^1$ compactifications. The colored arrow involves the $\mathbb{Z}_2$ twist on the charge lattice (denoted by R), $(-1)^F$ twist, and half shift on $S^1$ (denoted by T). Regarding 9d non-supersymmetric theories, we have used notation in Hohn:2023auwDeFreitas:2024ztt. Note that $A_I$ is rank $17$ while $B_{IIa}$, $B_{IIb}$, and $B_{III}$ are rank $9$. The red and blue colors correspond to supersymmetric and non-supersymmetric theories, respectively.
• Figure 2: Folding from $A_{2n-1}$ to $C_n$
• Figure 3: A codimension two Gukov-Witten operator (black line) charged under a dual $(D-2)$-form symmetry generates a holonomy along the transverse $S^1$ direction. The Gukov-Witten operator can be viewed as an insertion of the probe vortex with the holonomy. Consequently, this operator can end at the dynamical $(D-3)$-brane(Alice string/twisted vortex) represented by the red dot. The charge of the vortex is measured by the homotopy group $\pi_0$ or the bordism group $\Omega_1$.