Non-Geometric Vacua of the $\mathbf{\text{Spin}(32)/\mathbb Z_2}$ Heterotic String and Little String Theories

We study a class of 6d $\mathcal{N}=(1,0)$ non-geometric vacua of the $\text{Spin}(32)/\mathbb Z_2$ heterotic string which can be understood as fibrations of genus-two curves over a complex one-dimensional base. The 6d $\mathcal{N}=(1,0)$ theories living on the defects that arise when the genus-two fiber degenerates at a point of the base are analyzed by dualizing to F-theory on elliptic K3-fibered non-compact Calabi-Yau threefolds. We consider all possible degenerations of genus-two curves and systematically attempt to resolve the singularities of the dual threefolds. As in the analogous non-geometric vacua of the $E_8\times E_8$ heterotic string, we find that many of the resulting dual threefolds contain singularities which do not admit a crepant resolution. When the singularities can be resolved crepantly, we determine the emerging effective theories which turn out to be little string theories at a generic point on their tensor branch. We also observe a form of duality in which theories living on distinct defects are the same.