Graph calculus and the disconnected-boundary Schwinger-Dyson equations of quartic tensor field theories

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.