The full Ward-Takahashi Identity for colored tensor models

Colored tensor models (CTM) is a random geometrical approach to quantum gravity. We scrutinize the structure of the connected correlation functions of general CTM-interactions and organize them by boundaries of Feynman graphs. For rank-$D$ interactions including, but not restricted to, all melonic $\varphi^4$-vertices---to wit, solely those quartic vertices that can lead to dominant spherical contributions in the large-$N$ expansion---the aforementioned boundary graphs are shown to be precisely all (possibly disconnected) vertex-bipartite regularly edge-$D$-colored graphs. The concept of CTM-compatible boundary-graph automorphism is introduced and an auxiliary graph calculus is developed. With the aid of these constructs, certain $\mathrm U(\infty)$-invariance of the path integral measure is fully exploited in order to derive a strong Ward-Takahashi Identity for CTMs with a symmetry-breaking kinetic term. For the rank-$3$ $\varphi^4$-theory, we get the exact integral-like equation for the 2-point function. Similarly, exact equations for higher multipoint functions can be readily obtained departing from this full Ward-Takahashi identity. Our results hold for some Group Field Theories as well. Altogether, our non-perturbative approach trades some graph theoretical methods for analytical ones. We believe that these tools can be extended to tensorial SYK-models.