String islands, discrete theta angles and the 6D $\mathcal{N} = (1,1)$ string landscape

TL;DR

This work completes a perturbative classification of 6D $ obreak N=(1,1)$ string vacua by proving a one-to-one correspondence between Type II asymmetric orbifolds on $T^4$ and holomorphic orbifolds of chiral SCFTs with $c=12$ (and, correspondingly, $c=24$ for heterotic). It constructs eight inequivalent theories, including four RR-crystalline orbifolds and four RR-quasicrystalline islands, with explicit data for $n eq 2,3,4,6,8,10,12$, and shows that discrete theta angles occur for $n=5$ and $n=8$, thereby modifying both string and particle charge lattices and yielding BPS-incomplete spectra that violate the lattice weak gravity conjecture in a controlled way. The islands are connected, via circle compactification, to 5D quasicrystalline compactifications through a Sen–Vafa-type duality, providing a bridge between perturbative constructions and exotic nonperturbative frameworks. The results confirm the predicted landscape of perturbative 6D theories, elucidate the role of discrete theta angles in shaping charge lattices, and reveal rich dualities with higher-dimensional compactifications. These advances sharpen our understanding of higher-dimensional string universes with minimal moduli and highlight new avenues for exploring nonperturbative constraints in quantum gravity.

Abstract

The complete classification of the landscape of 6D $\mathcal{N} = (1,1)$ string vacua remains an open problem. In this work we prove a classification theorem for 6D $\mathcal{N} = (1,1)$ asymmetric orbifolds utilizing a correspondence with orbifolds of chiral 2D SCFTs with $c= 24$ (or $c= 12$). Interestingly, this class of theories can give rise to 6D vacua in which the only massless degrees of freedom reside in the gravity multiplet, with no moduli other than the dilaton, thus corresponding to truly isolated vacua, called string islands. It is expected that there exist five new type II islands with as-yet-unknown constructions. In this work we construct them all using asymmetric $\mathbb{Z}_n$-orbifolds of Type II on $T^4$ with $n = 5,8,10,12$. We show that the cases $n = 5,8$ admit non-trivial discrete theta angles which have important consequences for both the string and particle charge lattices. In fact they provide examples of BPS-incompleteness and the strongest failure of the lattice weak gravity conjecture. Our work is expected to finalize our understanding of all perturbative 6D $\mathcal{N} = (1,1)$ theories.

Figures (1)

• Figure 1: Correspondence between asymmetric orbifolds of the type II string on $T^4$ and the 2D superconformal field theory $F_{24}$ comprising 24 fermions and a current specified by the given Kac-Moody algebra. The rank of the type II theory is equal to the rank of the corresponding current minus 4, hence the theories in the lower row have rank 0 and are accordingly referred to as string islands following Dabholkar:1998kv. The symbol $\hat{A}_{m,n}$ denotes the Kac-Moody algebra for $SU(m+1)$ at level $n$, and likewise, $\hat{B}_{n,m}$, $\hat{C}_{m,n}$ and $\hat{G}_{2,m}$ refer to $SO(2m+1)$, $Sp(m)$ and $G_2$.