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

A bstract The complete classification of the landscape of 6D $$ \mathcal{N} $$ N = (1) string vacua remains an open problem. In this work we prove a classification theorem for 6D $$ \mathcal{N} $$ N = (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 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 $$ \mathcal{N} $$ N = (1) theories.