  We explore the dynamics of a simple class of two-dimensional models with
$(0,1)$ supersymmetry, namely sigma-models with target $S^3$ and the minimal
possible set of fields. For any nonzero value of the Wess--Zumino coupling $k$,
we describe a superconformal fixed point to which we conjecture that the model
flows in the infrared. For $k=0$, we conjecture that the model spontaneously
breaks supersymmetry. We further explore the question of whether this model can
be continuously connected to one that spontaneously breaks supersymmetry by
"flowing up and down the renormalization group trajectories," in a sense that
we describe. We show that this is possible if $k$ is a multiple of 24, or
equivalently if the target space with its $B$-field is the boundary of a
"string manifold." The mathematical theory of "topological modular forms"
suggests that this condition is necessary as well as sufficient.
