Non-abelian asymmetric orbifolds with vanishing one-loop vacuum energy

We present a partial classification of four-dimensional, non-supersymmetric Type II toroidal orbifolds with non-abelian point groups of rotations, and vanishing vacuum energy at one loop in string perturbation theory. The classification is complete within a class of such orbifolds with the restriction that the point group only acts on a $T^5$ inside the full internal $T^6$. By studying their decompactification limits along the leftover $S^1$, we see that the 17 solutions we find can alternatively be obtained as Scherk–Schwarz compactifications of some parent asymmetric orbifold. Along the way, to ensure the absence of anomalies, we compute bordism groups ${\mathrm{\Omega }}_3^{Spin(BG)}$ for a variety of non-abelian crystallographic groups $G$.