  In various dimensions, we can sometimes compute observables of interacting
conformal field theories (CFTs) that are connected to free theories via the
renormalization group (RG) flow by computing protected quantities in the free
theories. On the other hand, in two dimensions, it is often possible to
algebraically construct observables of interacting CFTs using free fields
without the need to explicitly construct an underlying RG flow. In this note,
we begin to extend this idea to higher dimensions by showing that one can
compute certain observables of an infinite set of unitary strongly interacting
four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) by
performing simple calculations involving sets of non-unitary free
four-dimensional hypermultiplets. These free fields are distant cousins of the
Majorana fermion underlying the two-dimensional Ising model and are not
obviously connected to our interacting theories via an RG flow. Rather
surprisingly, this construction gives us Lagrangians for particular observables
in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity
while preserving the full $\mathcal{N}=2$ superconformal algebra. As a
byproduct, we find relations between characters in unitary and non-unitary
affine Kac-Moody algebras. We conclude by commenting on possible
generalizations of our construction.
