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The Full Ward-Takahashi Identity for Colored Tensor Models

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A Correction to this article was published on 01 February 2020

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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 \({\varphi^4}\) -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 U (∞)-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.

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Change history

  • 01 February 2020

    The author wishes to rectify some oversights and, mainly, to amend an erroneous term.

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  1. Mathematisches Institut, Westfälische Wilhelms-Universität, Einsteinstraße 62, 48149, Münster, Germany

    Carlos I. Pérez-Sánchez

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Correspondence to Carlos I. Pérez-Sánchez.

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Communicated by M. Salmhofer

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Pérez-Sánchez, C.I. The Full Ward-Takahashi Identity for Colored Tensor Models. Commun. Math. Phys. 358, 589–632 (2018). https://doi.org/10.1007/s00220-018-3103-2

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