We consider a 4d scalar field coupled to large $N$ free or critical $O(N)$ vector models, either bosonic or fermionic, on a 3d boundary. We compute the $\beta$ function of the classically marginal bulk/boundary interaction at the first non-trivial order in the large $N$ expansion and exactly in the coupling. Starting with the free (critical) vector model at weak coupling, we find a fixed point at infinite coupling in which the boundary theory is the critical (free) vector model and the bulk decouples. We show that a strong/weak duality relates one description of the renormalization group flow to another one in which the free and the critical vector models are exchanged. We then consider the theory with an additional Maxwell field in the bulk, which also gives decoupling limits with gauged vector models on the boundary.

中文翻译：

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边界处的 3d 大 $N$ 矢量模型

我们考虑在 3d 边界上耦合到大 $N$ 或临界 $O(N)$ 向量模型（玻色子或费米子）的 4d 标量场。我们计算了经典边际体积/边界相互作用的 $\beta$ 函数，在大 $N$ 扩展中的第一个非平凡的顺序上，并且恰好在耦合中。从弱耦合下的自由（临界）矢量模型开始，我们在无限耦合处找到一个不动点，其中边界理论是临界（自由）矢量模型和体解耦。我们表明，强/弱对偶性将重整化群流的一种描述与另一种交换自由和临界向量模型的描述相关联。然后，我们考虑在体中具有额外麦克斯韦场的理论，这也给出了边界上测量矢量模型的解耦限制。