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Intermediate Field Representation for Positive Matrix and Tensor Interactions

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Abstract

In this paper we introduce an intermediate field representation for random matrices and random tensors with positive (stable) interactions of degree higher than 4. This representation respects the symmetry axis responsible for positivity. It is non-perturbative and allows to prove that such models are Borel–Le Roy summable of the appropriate order in their coupling constant. However, we have not been able yet to associate a convergent loop vertex expansion with this representation; hence, our Borel summability result is not of the optimal expected form when the size N of the matrix or of the tensor tends to infinity.

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Notes

  1. Note added: Between submission and acceptance of this paper this problem has been solved: The Borel radius has been indeed proved in the matrix case not to shrink as \(N \rightarrow \infty \) [72].

  2. We can extend these formulas to the case \(C=0\) by defining \(\mathrm{d}\mu \) in this case to be the Dirac measure at the origin.

  3. We decided to put the N dependence on the Borel radius and none on \(\epsilon \). Other choices could include an N dependence on \(\epsilon \) but would lead to contour integrals no longer defined in the large N limit.

  4. This is the crucial property from the point of view of gravity quantization since it allows to associate with the dual space a canonical discretized Einstein–Hilbert action.

  5. Beware, however, that the optimal scaling s is not known for general tensor invariants [70].

  6. We use a simplifying notation to have clearer expressions. The sum over a color set J here means a sum for variables indexed with each color in J.

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Authors and Affiliations

  1. Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, 91405, Orsay Cedex, France

    Luca Lionni & Vincent Rivasseau

  2. LIPN, UMR CNRS 7030, Institut Galilée, Université Paris 13, Sorbonne Paris Cité, 99, avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Luca Lionni

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  1. Luca Lionni
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Correspondence to Vincent Rivasseau.

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Lionni, L., Rivasseau, V. Intermediate Field Representation for Positive Matrix and Tensor Interactions. Ann. Henri Poincaré 20, 3265–3311 (2019). https://doi.org/10.1007/s00023-019-00833-z

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