One Loop to Rule Them All: Eight and Nine Dimensional String Vacua from Junctions

TL;DR

The paper develops a comprehensive junction-based framework to classify all known 8d ${\cal N}=1$ string vacua, incorporating O7$^+$ planes to realize ${\mathfrak{sp}}_n$ algebras and their center symmetries. By analyzing string and 5-brane junctions, loop junctions, and fractional null junctions, it derives the global gauge-group topology, connects to heterotic/CHL dual descriptions via Narain and Mikhailov lattices, and extends to 9d uplifts through affine 7-brane stacks. It further introduces a rank $(2,2)$ momentum-lattice description and provides explicit maximally-enhanced examples across the three 8d branches, matching CHL and heterotic results. The work reveals two affine-driven branches in 9d rank $(1,1)$ vacua and demonstrates that the junction approach yields a unified, geometry-based map between 8d vacua and their 9d uplifts, with significant Swampland implications. Overall, junctions with O7$^+$ planes unify the realization of gauge algebras and their higher-form symmetries across 8d and 9d, offering a robust tool for exploring nonperturbative string vacua.

Abstract

String and 5-brane junctions are shown to succinctly classify all known 8d ${\cal N}$=1 string vacua. This requires an extension of the description for ordinary $[p,q]$-7-branes to consistently include O7$^+$-planes, which then naturally encodes the dynamics of $\mathfrak{sp}_n$ gauge algebras, including their higher-form center symmetries. Central to this analysis are loop junctions, i.e., strings/5-branes which encircle stacks of 7-branes and O7$^+$'s. Loop junctions further signal the appearance of affine symmetries of emergent 9d descriptions at the 8d moduli space's boundaries. Such limits reproduce all 9d string vacua, including the two disconnected rank (1,1) moduli components.

Figures (8)

• Figure 1: Strings and 5-branes, which are represented as lines in the perpendicular plane, form junctions, where the $\left( pq \right)$-charge at each vertex is conserved (left). In the presence of 7-branes, they undergo monodromy transformations \ref{['eq:monodromy_on_prongs']} when they cross a branch cut (middle). By a Hanany--Witten transition, the same junction can be represented as having a prong on the 7-brane (right).
• Figure 2: A loop junction $\boldsymbol\ell_{(r,s)}$ around a collection of 7-branes with overall monodromy $M$. The asymptotic charge $\left( pq \right) = \left( r's' \right) - \left( rs \right) = (M - \mathbb{1}) \left( rs \right)$ is in general non-zero.
• Figure 3: The self-pairing of a 3-pronged junction (left) can be separated into contributions from the ends on 7-branes and the vertex, see \ref{['eq:self_pairing_3-pronged']}. When there are no prongs ending on 7-branes, such as for loop juncions (right), the only contribution is that of the vertex.
• Figure 4: Construction of extended weight junctions. Since the $\left( rs \right)$-charges that appear in the loop are in general fractional, the prongs ending on the 7-branes after pulling the loop across also have fractional coefficients.
• Figure 5: Root junctions of $\mathfrak{sp}$ algebra (the double arrow denotes the factor of 2 required by evenness on O7$^+$).