$O(N)$ Random Tensor Models

TL;DR

The paper develops real tensor models with $O(N)^{\otimes3}$ invariance, showing a robust $1/N$ expansion for general interactions and focusing on a quartic model that combines tetrahedral and pillow interactions. Leading graphs are melonic, as in MO/$U(N)^{\otimes3}$ models, while next-to-leading graphs are obtained by melonic 2-point insertions into a finite set of core graphs, enabling a precise analytic combinatorics treatment. The authors derive generating functions, diagrammatic equations, and a singularity analysis that yield the critical curve $g_c(\mu)$ and establish LO and NLO critical exponents $\gamma_{LO}=1/2$ and $\gamma_{NLO}=3/2$, indicating universality with known tensor models. These results pave the way for double-scaling limits and broader explorations of tensorial group field theories in an enlarged interaction space.

Abstract

We define in this paper a class of three indices tensor models, endowed with $O(N)^{\otimes 3}$ invariance ($N$ being the size of the tensor). This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model (and hence of the colored model) and the $U(N)$ invariant models. We first exhibit the existence of a large $N$ expansion for such a model with general interactions. We then focus on the quartic model and we identify the leading and next-to-leading order (NLO) graphs of the large $N$ expansion. Finally, we prove the existence of a critical regime and we compute the critical exponents, both at leading order and at NLO. This is achieved through the use of various analytic combinatorics techniques.

Figures (14)

• Figure 1: The two- and four-valent bubbles/vertices of the model.
• Figure 2: The unique MO interaction vertex.
• Figure 3: A Feynman graph of the $\mathrm{O}(N)^{\otimes 3}$ model which is neither MO nor bipartite.
• Figure 4: Two vacuum (and melonic) graphs.
• Figure 5: Jackets associated to the tensor graphs of Figure \ref{['ex_ampl']}. On the right-hand-side, two connected components are generated by the removal of the lines of color $\ell$.

Theorems & Definitions (33)

• Proposition 1
• proof
• Definition 1
• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Definition 2