论文标题
大型$ r $ - 电荷运营商在保形歧管上的过渡
Transition of Large $R$-Charge Operators on a Conformal Manifold
论文作者
论文摘要
我们研究了在保形的歧管上以大$ r $ charge的阶段之间的过渡。这些阶段的特征是最低运算符尺寸$δ（q_r）$的行为，用于固定和大$ r $ - charge $ q_r $。例如，我们将重点放在$ d = 3 $，$ \ mathcal {n} = 2 $ wess-zumino型号，带有立方超级电位$ w = xyz+\ frac {6}（x^+y^3+y^3+z^3）$，并计算使用$ -Expans $ thrim-expans $ thrim-expans $ - 在以下两个限制中，（领先顺序）结果被证明为\ begin {equation*} δ（q_r，τ）= \ begin {case} \ left（\ text {bps bond} \右）\ left [1+o（ε| |^|^|^2q_r）\ right]，＆q_r \ ll \ ll \ left \ left \ {\ frac {1} {1}ε，\，\，\，\ frac，\ frac {1}} \ frac {9} {8} \ left（\ frac {ε| |^|^2} {2+ | |τ|^2} \ right）^{\ frac {1} {d-1} {d-1}} q_r^ \ left [1+o \ left（\ left（ε|^|^2q_r \ right）^{ - \ frac {2} {d-1}}}}}} \ right）\ right]，]， ＆q_r \ gg \ left \ {\ frac {1}ε，\，\ frac {1} {ε| |^2} \ right \} \ end {cases} \ end {equation*}，它导致我们进入双尺度参数，$ε|τ|^2q_r $，它在“近bps相”（$δ（q）\ sim q $）和“ superfluid阶段”（$δ（q）\ sim q^q^Q^{d/d/d/d/d/$ r之间）插入了“近bps相”（$δ（q）\ sim q $）。这种平稳的过渡发生在$τ= 0 $附近，是模量空间的存在和$τ= 0 $的无限手性环的存在的大$ r $ charge表现。我们还认为，这种行为可以扩展到最小修改的三个维度，因此我们得出的结论是，$δ（q_r，τ）$经历了左右的平滑过渡，围绕$ q_r \ sim 1/|τ|^2 $。此外，由于模型的双重性，我们发现$δ（q_r，τ）$的一阶相变是$τ$的函数。我们还将结果的适用性评论至小$ r $ charge。
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 $Δ(Q_R)$ 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 $W=XYZ+\fracτ{6}(X^3+Y^3+Z^3)$, and compute $Δ(Q_R,τ)$ using the $ε$-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, $ε|τ|^2Q_R$, which interpolates between the "near-BPS phase" ($Δ(Q)\sim Q$) and the "superfluid phase" ($Δ(Q)\sim Q^{D/(D-1)}$) at large $R$-charge. This smooth transition, happening near $τ=0$, is a large-$R$-charge manifestation of the existence of a moduli space and an infinite chiral ring at $τ=0$. We also argue that this behavior can be extended to three dimensions with minimal modifications, and so we conclude that $Δ(Q_R,τ)$ experiences a smooth transition around $Q_R\sim 1/|τ|^2$. Additionally, we find a first-order phase transition for $Δ(Q_R,τ)$ as a function of $τ$, as a consequence of the duality of the model. We also comment on the applicability of our result down to small $R$-charge.
