Transition of large R-charge operators on a conformal manifold

We study the transition between phases at large $R$-charge on a conformal manifold. These are characterized by behaviour of lowest operator dimension $\Delta(Q_R)$ for fixed and $Q_R$. focus, as an example, $D=3$, $\mathcal{N}=2$ Wess-Zumino model with cubic superpotential $W=XYZ+\frac{\tau}{6}(X^3+Y^3+Z^3)$, compute $\Delta(Q_R,\tau)$ using $\epsilon$-expansion in three interesting limits. In two these limits (leading order) result turns out to be \begin{equation*} \Delta(Q_R,\tau)= \begin{cases} \left(\text{BPS bound}\right)\left[1+O(\epsilon |\tau|^2Q_R)\right], & Q_R\ll \left\{ \frac{1}{\epsilon},\, \frac{1}{\epsilon|\tau|^2}\right\}\\ \frac{9}{8}\left(\frac{\epsilon|\tau|^2}{2+|\tau|^2}\right)^{\frac{1}{D-1}}Q_R^{\frac{D}{D-1}} \left[1+O\left(\left(\epsilon |\tau|^2Q_R\right)^{-\frac{2}{D-1}}\right)\right], Q_R\gg \frac{1}{\epsilon|\tau|^2}\right\} \end{cases} \end{equation*} which leads us double-scaling parameter, $\epsilon |\tau|^2Q_R$, interpolates "near-BPS phase" ($\Delta(Q)\sim Q$) "superfluid Q^{D/(D-1)}$) $R$-charge. This smooth transition, happening near $\tau=0$, is large-$R$-charge manifestation existence moduli space infinite chiral ring $\tau=0$. also argue that this behavior can extended dimensions minimal modifications, so we conclude experiences around $Q_R\sim 1/|\tau|^2$. Additionally, find first-order phase function $\tau$, consequence duality model. comment applicability our down small