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Flowing in discrete gravity models and Ward identities: a review

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Abstract

Ward–Takahashi identities are nonperturbative relations between correlation functions and arising from symmetries in quantum and statistical fields theories, as Noether currents conservation for classical theories. Since their historical origin, these identities were considered to prove the exact relation between counter-terms to all order of the perturbative expansion. Recently they have been considered in relation with nonperturbative renormalization group aspects for some classes of quantum field theories namely tensorial group field theories and matrix models, both characterized by a specific non-locality in their interactions, and expected to provide discrete models for quantum gravity. In this review, we summarize the state of the art, focusing on the conceptual aspects rather than technical subtleties, and provide a unified reflection on this novel and promising way of investigation. We attached great importance to the pedagogy and the self-consistency of the presentation.

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Notes

  1. Note that we missed a factor 2 in the computation of [127, 128, 135].

    $$\begin{aligned} \frac{\hbox {d}}{\hbox {d}p^2}\mathcal {A}_{2,p}\bigg \vert _{p=0}=-\frac{1}{Z^2k^2}\frac{\pi ^2}{1+\bar{m}^2}\left( 1+\frac{1}{1+\bar{m}^2} \right) \,.\ \end{aligned}$$
    (37)

  2. Note that however these corrections do not modify the result significantly.

  3. This argument, obviously is subordinated to our approximations.

  4. This condition may be refined, see [128], but this point has no consequence on our discussion.

  5. See [129,130,131,132,133] and references therein for an extended discussion, showing how this definition work in practical contexts, especially in the context of matrix field theory to define counter-terms for renormalization.

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  1. Faculté des Sciences et Techniques (ICMPA-UNESCO Chair), Université d’Abomey-Calavi, 072 BP 50, Cotonou, Benin

    Dine Ousmane Samary & Ezinvi Baloïtcha

  2. Commissariat à l’Énergie Atomique (CEA, LIST), 8 Avenue de la Vauve, 91120, Palaiseau, France

    Dine Ousmane Samary & Vincent Lahoche

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Ousmane Samary, D., Lahoche, V. & Baloïtcha, E. Flowing in discrete gravity models and Ward identities: a review. Eur. Phys. J. Plus 136, 982 (2021). https://doi.org/10.1140/epjp/s13360-021-01823-z

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