Motivated by applications to critical phenomena and open theoretical questions, we study conformal field theories with $O(m)\times O(n)$ global symmetry in $d=3$ spacetime dimensions. We use both analytic and numerical bootstrap techniques. Using the analytic bootstrap, we calculate anomalous dimensions and OPE coefficients as power series in $\varepsilon=4-d$ and in $1/n$, with a method that generalizes to arbitrary global symmetry. Whenever comparison is possible, our results agree with earlier results obtained with diagrammatic methods in the literature. Using the numerical bootstrap, we obtain a wide variety of operator dimension bounds, and we find several islands (isolated allowed regions) in parameter space for $O(2)\times O(n)$ theories for various values of $n$. Some of these islands can be attributed to fixed points predicted by perturbative methods like the $\varepsilon$ and large-$n$ expansions, while others appear to arise due to fixed points that have been claimed to exist in resummations of perturbative beta functions.

中文翻译：

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3D中具有$ O（m）\ times O（n）$全局对称的CFT的解析和数值引导

受到对关键现象和开放理论问题的应用的激励，我们研究了在$ d = 3 $时空维度中具有$ O（m）\ timesO（n）$全局对称性的共形场论。我们同时使用解析和数值引导技术。使用解析引导程序，我们使用归纳为任意全局对称性的方法，将异常尺寸和OPE系数计算为$ \ varepsilon = 4-d $和$ 1 / n $的幂级数。只要有可能进行比较，我们的结果就与文献中使用图解法获得的早期结果一致。使用数值引导程序，我们获得了多种运算符维数边界，并且在$ O（2）\ O（n）$理论的参数空间中，对于$ n $的各种值，我们发现了几个孤岛（隔离的允许区域）。