New Supersymmetric String Moduli Spaces from Frozen Singularities

The current classification of $\mathcal{N}=1$ string theories in eight and seven dimensions is completely captured by K3 surfaces with F-Theory or M-Theory frozen singularities. In this note we show that there are inequivalent ways of freezing certain collections of singularities which have the same ADE type, and so there are more connected components in the moduli space than previously thought; namely, one more in eight dimensions and three more in seven dimensions. We argue that the new component in eight dimensions decompactifies to a string theory in nine dimensions with rank 1 gauge group, which has been so far unidentified. Constructing and studying the stringy descriptions corresponding to these moduli spaces is the subject of a companion paper.