  We study the one-dimensional complex conformal manifold that controls the
infrared dynamics of a three-dimensional $\mathcal{N}=2$ supersymmetric theory
of three chiral superfields with a cubic superpotential. Two special points on
this conformal manifold are the well-known XYZ model and three decoupled copies
of the critical Wess-Zumino model. The conformal manifold enjoys a discrete
duality group isomorphic to $S_4$ and can be thought of as an orbifold of
$\mathbf{CP}^1$. We use the $4-\varepsilon$ expansion and the numerical
conformal bootstrap to calculate the spectrum of conformal dimensions of
low-lying operators and their OPE coefficients, and find a very good
quantitative agreement between the two approaches.
