  Spectral flow, spacetime supersymmetry, topological twists, chiral primaries
related to marginal deformations, mirror symmetry: these are important
consequences of the worldsheet N=2 superconformal symmetry of strings on
Calabi-Yau manifolds. To various degrees of certainty, these features were also
established when the target is either 7d or 8d with exceptional holonomy G$_2$
or Spin(7) respectively. We show that these are more than mere analogies. We
exhibit an underlying symmetry SW(3/2,2) making a bridge between the latter
cases and K3 target spaces. Reviewing unitary representations of SW(3/2,2)
leads us to speculate on further roles of this algebra in string theory
compactifications and on the existence of topologically twisted versions of
SW(3/2,2) theories.
