An infinite swampland of U(1) charge spectra in 6D supergravity theories

TL;DR

This paper analyzes anomaly cancellation constraints for six-dimensional N=1 supergravity theories with a single abelian U(1) factor, revealing an infinite swampland of massless charge spectra that satisfy known consistency conditions but cannot be realized by F-theory or other established string constructions, especially when tensor multiplets are present or charges exceed |q|=2. It shows a tight match between anomaly constraints and F-theory realizations in the simplest case (T=0, |q|≤2), and demonstrates that every anomaly-free U(1) model in this regime can be unHiggsed to an SU(2) theory, highlighting a strong link between abelian and nonabelian spectra. Beyond this regime, the authors construct infinite families of anomaly-free spectra, develop continuous and discrete counting arguments for the growth of solutions, and classify models with higher charges up to |q|≤3, identifying numerous exchanges and unHiggsing pathways to nonabelian theories. They discuss F-theory bounds, such as Kodaira constraints and Morrison–Park constructions, and note that exotic non-UFD realizations can violate these bounds, suggesting deeper structure behind the swampland and pointing to the need for new UV consistency conditions. Overall, the work clarifies how abelian charge spectra relate to nonabelian Higgsings, enumerates extensive swampland sectors, and frames future directions for tightening F-theory realizations and quantum-consistency criteria in 6D gravity with abelian factors.

Abstract

We analyze the anomaly constraints on 6D supergravity theories with a single abelian U(1) gauge factor. For theories with charges restricted to $q = \pm1, \pm2$ and no tensor multiplets, anomaly-free models match those models that can be realized from F-theory compactifications almost perfectly. For theories with tensor multiplets or with larger charges, the F-theory constraints are less well understood. We show, however, that there is an infinite class of distinct massless charge spectra in the "swampland" of theories that satisfy all known quantum consistency conditions but do not admit a realization through F-theory or any other known approach to string compactification. We also compare the spectra of charged matter in abelian theories with those that can be realized from breaking nonabelian SU(2) and higher rank gauge symmetries.

Figures (2)

• Figure 1: An integer tuple $(a, b, c, d)$ such that there exists an equilateral triangle of side length $c$ with an integral cevian of length $d / 2$ dividing the side into integral parts, $a + b = c$. Such tuples produce anomaly-free models of the form \ref{['eq:infinite-4family']}.
• Figure 2: The prediction for the distribution of values of $x = (q^4 + r^4) / (q^2 + r^2)^2$ compared with numerical values for all $(q, r)$ with $q^2 + r^2 \le 10\,000$, where numerical values are averaged over bins of size $1 / 20$ (left) and $1 / 200$ (right). Irregularities in the right diagram indicate number-theoretical patterns that become relevant for bins smaller than $1 / \sqrt{B}$, such as the large peak at $x = 0.68$, as mentioned in the text.