We study finite group actions that are parametrised by coloured graphs, as
the basis of the graph calculus. In this setting, a derivative with respect to
a certain graph yields its respective group action. The graph calculus is built
on a suitable quotient of the monoid algebra $A[G]$ corresponding to a certain
function space $A$ and the free monoid $G$ in finitely many graph variables.
The largest section is dedicated solely to these algebraic structures, which,
although motivated by Tensor Field Theory (TFT), are introduced and dealt with
without reference to it. These abstract results are subsequently applied to a
TFT problem:
Tensor field theory focuses on quantum field theory aspects of random tensor
models, a quantum-gravity-motivated generalisation of random matrix models. The
correlation functions of complex tensor models have a rich combinatorial
structure: they are classified by boundary graphs that describe the geometry of
the boundary states. 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
use a graph calculus---where derivatives of graphs yield group actions by their
coloured automorphism---in order to report on the missing equations for
correlation functions with disconnected boundary, thus completing the
Schwinger-Dyson pyramid for quartic melonic ('pillow'-vertices) in arbitrary
rank. We hope that the present result sheds light on the non-perturbative
large-$N$ limit of tensor field theories. Moreover, we presume that it can be
interesting if one addresses the solvability of the theory by using methods
that generalise the topological recursion to the tensor field theory setting.