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 phases are characterized by the behaviour of the lowest operator dimension $\mathrm{\Delta }(Q_R)$ for fixed and large $R$-charge $Q_R$. We focus, as an example, on the $D=3$, $\mathcal{N}=2$ Wess-Zumino model with cubic superpotential $W=XYZ+\frac{\tau }{6}(X^3+Y^3+Z^3)$, and compute $\mathrm{\Delta }(Q_R,\tau )$ using the $ϵ$-expansion in three interesting limits. In two of these limits the (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 \left{ \frac{1}{\epsilon},, \frac{1}{\epsilon|\tau|^2}\right} \end{cases} \end{equation*} which leads us to the double-scaling parameter, $\epsilon |\tau|^2Q_R$,whichinterpolatesbetweenthe\mathrm{"}near-BPSphase\mathrm{"}($\Delta(Q)\sim Q$)andthe\mathrm{"}superfluidphase\mathrm{"}($\Delta(Q)\sim Q^{D/(D-1)}$)atlarge$R$-charge\mathrm{.}Thissmoothtransition,happeningnear$\tau=0$, is a large-$R$-charge manifestation of the existence of a moduli space and an infinite chiral ring at $\tau =0$. We also argue that this behavior can be extended to three dimensions with minimal modifications, and so we conclude that $\mathrm{\Delta }(Q_R,\tau )$ experiences a smooth transition around $Q_R\sim 1/|\tau|^2$\mathrm{.}Additionally,wefindafirst-orderphasetransitionfor$\Delta(Q_R,\tau)$asafunctionof$\tau$, as a consequence of the duality of the model. We also comment on the applicability of our result down to small $R$-charge.