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

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.