The full Ward-Takahashi Identity for colored tensor models

Abstract

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 ${\phi }^4$-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-$3$ ${\phi }^4$-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.

39 pages, TikZ-figures. v5: Amend typo in to eq. (51), and syntaxis of Lemma 4

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• Unitary symmetry constraints on tensorial group field theory renormalization group flow
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• On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra
• Pedagogical comments about nonperturbative Ward-constrained melonic renormalization group flow
• On the large N limit of the Schwinger-Dyson equation of tensor field theory
• Surgery in colored tensor models
• No Ward-Takahashi identity violation for an Abelian tensorial group field theories with closure constraint
• A Solvable Tensor Field Theory
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Correlation functions of $\mathrm{U}(N)$-tensor models and their Schwinger-Dyson equations
• Comment on "The phase diagram of the multi-matrix model with ABAB-interaction from functional renormalization"
• Graph calculus and the disconnected-boundary Schwinger-Dyson equations of quartic tensor field theories