Transition of Large $R$-Charge Operators on a Conformal Manifold

Abstract

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 $\hat{I}”(Q_R)$ 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 $W=XYZ+\frac{\ddot{I}}{„}6(X^3+Y^3+Z^3)$, and compute $\hat{I}”(Q_R,\ddot{I}„)$ using the $\hat{I}\mu$-expansion in three interesting limits. In two of these limits the (leading order) result turns out to be \begin{equation*} Δ(Q_R,τ)= \begin{cases} \left(\text{BPS bound}\right)\left[1+O(ε|τ|^2Q_R)\right], & Q_R\ll \left\{ \frac{1}ε,\, \frac{1}{ε|τ|^2}\right\}\\ \frac{9}{8}\left(\frac{ε|τ|^2}{2+|τ|^2}\right)^{\frac{1}{D-1}}Q_R^{\frac{D}{D-1}} \left[1+O\left(\left(ε|τ|^2Q_R\right)^{-\frac{2}{D-1}}\right)\right], & Q_R\gg \left\{ \frac{1}ε,\, \frac{1}{ε|τ|^2}\right\} \end{cases} \end{equation*} which leads us to the double-scaling parameter, $\hat{I}\mu |\ddot{I}„{|}^2{Q}_R$, which interpolates between the "near-BPS phase" ($\hat{I}”(Q)\sim Q$) and the "superfluid phase" ($\hat{I}”(Q)\sim Q^{D/(D-1)}$) at large $R$-charge. This smooth transition, happening near $\ddot{I}„=0$, is a large-$R$-charge manifestation of the existence of a moduli space and an infinite chiral ring at $\ddot{I}„=0$. We also argue that this behavior can be extended to three dimensions with minimal modifications, and so we conclude that $\hat{I}”(Q_R,\ddot{I}„)$ experiences a smooth transition around $Q_R\sim 1/|\ddot{I}„{|}^2$. Additionally, we find a first-order phase transition for $\hat{I}”(Q_R,\ddot{I}„)$ as a function of $\ddot{I}„$, as a consequence of the duality of the model. We also comment on the applicability of our result down to small $R$-charge.