Séminaire Physique mathématique ICJ

The large N melonic limit of O(N) tensor models

par Sylvain Carrozza (Perimeter Institute)

Fokko du Cloux (Institut Camille Jordan)

Fokko du Cloux

Institut Camille Jordan

Université Lyon 1, Bât. Braconnier, 21 av. Claude Bernard, 69100 Villeurbanne
Tensor models are generalizations of matrix models which describe the dynamics of fields with r > 2 indices. As discovered some years ago, they enjoy a large N expansion which is (perhaps surprisingly) much simpler than the large N expansion of matrix models. It is dominated by the so-called "melonic" family of Feynman diagrams, which can sometimes be resumed explicitly. Following Witten and Klebanov-Tarnopolsky, this has recently led to the definition of solvable strongly coupled quantum theories, which reproduce the main properties of the celebrated Sachdev-Ye-Kitaev condensed matter models. Most of the literature on tensor models focuses on tensor degrees of freedom transforming under r independent copies of a symmetry group G, one for each index (for definiteness, I will focus on r=3 and G=O(N)). This large symmetry plays a crucial role in the analysis of the 1/N expansion, so much so that it was generally believed to be essential to its existence. After summarizing these results, I will outline the recent proof that irreducible O(N) tensors (e.g. symmetric traceless ones) also support a melonic 1/N expansion. This in particular confirms a conjecture recently put forward by Klebanov and Tarnopolsky, which had only been checked numerically up to order 8 in perturbative expansion.
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