Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing equations for all correlation functions with disconnected boundary, thus completing the Schwinger-Dyson pyramid for quartic melonic (`pillow'-vertices) models in arbitrary rank. We first study finite group actions that are parametrised by graphs and build the graph calculus on a suitable quotient of the monoid algebra $A[G]$ corresponding to a certain function space $A$ and to the free monoid $G$ in finitely many graph variables; a derivative of an element of $A[G]$ with respect to a graph yields its corresponding group action on $A$. The present result and the graph calculus have three potential applications: the non-perturbative large-$N$ limit of tensor field theories, the solvability of the theory by using methods that generalise the topological recursion to the TFT setting and the study of `higher dimensional maps' via Tutte-like equations. In fact, we also offer a term-by-term comparison between Tutte equations and the present Schwinger-Dyson equations.

中文翻译：

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图微积分和四次张量场理论的不连通边界施温格-戴森方程

张量场论 (TFT) 侧重于随机张量模型的量子场论方面，这是随机矩阵模型的量子引力驱动泛化。TFT 相关函数已被证明可以通过描述边界状态几何形状的图进行分类，即所谓的边界图。尽管相关函数本身是连接的，但这些图可以断开连接。在最近的一项工作中，获得了任意连接边界的 Schwinger-Dyson 方程。在这里，我们引入了多变量图演算，以推导出所有边界不连接的相关函数的缺失方程，从而完成任意等级的四次瓜（“枕头”顶点）模型的施温格-戴森金字塔。我们首先研究由图参数化的有限群动​​作，并在对应于某个函数空间 $A$ 的幺半群代数 $A[G]$ 的合适商上建立图演算，该商对应于某个函数空间 $A$ 和有限多自由幺半群 $G$图变量；$A[G]$ 的元素相对于图的导数产生其对 $A$ 的相应组动作。目前的结果和图演算具有三个潜在的应用：张量场理论的非微扰大 $N$ 极限，通过使用将拓扑递归推广到 TFT 设置的方法的理论可解性以及对“更高”的研究。维映射'通过类似 Tutte 的方程。事实上，我们还提供了 Tutte 方程和当前 Schwinger-Dyson 方程之间的逐项比较。