String Universality and Non-Simply-Connected Gauge Groups in 8d

TL;DR

We address which non-simply-connected gauge groups $G/Z$ can arise in 8d $\mathcal{N}=1$ supergravity by deriving a mixed anomaly between the center 1-form symmetry and the higher-form gauge sector. The key tool is the $B_4$-coupling to instanton densities, yielding an anomaly phase $\mathcal{A}(b_4, C_2^{(i)}) = \sum_i m_i \alpha_{G_i} \int_{M_8} b_4 \cup \mathfrak{P}(C_2^{(i)})$ that must be integral, which constrains feasible $Z \subset Z(G)$. In rank-18 theories with $m_i=1$, this anomaly condition reproduces the geometric Miranda–Persson constraints from elliptic K3/F-theory, allowing only specific quotients such as $SU(7)^3/\mathbb{Z}_7$ and $SU(8)^2 \times SU(4) \times SU(2)/\mathbb{Z}_8$ and ruling out others; at $m=1$ and rank $\le 18$ a finite list of simple quotients is allowed, with additional options at $m=2$ reflecting CHL/heterotic constructions. Together with Swampland bounds, these results provide a physically grounded account for why many seemingly consistent 8d gauge structures have no string realization, and offer concrete predictions for yet-unexplored string models. The analysis is currently a necessary but not sufficient criterion, motivating further work to incorporate $U(1)$ sectors and other symmetry structures toward a full String Universality statement in eight dimensions.

Abstract

We present a consistency condition for 8d ${\cal N} = 1$ supergravity theories with non-trivial global structure $G/Z$ for the non-Abelian gauge group, based on an anomaly involving the $Z$ 1-form center symmetry. The interplay with other Swampland criteria identifies the majority of 8d theories with gauge group $G/Z$, which have no string theory realization, as inconsistent quantum theories when coupled to gravity. While this condition is equivalent to geometric properties of elliptic K3 surfaces in F-theory compactifications, it constrains the unexplored landscape of gauge groups in other 8d string models.