The $1/N$ expansion of tensor models with two symmetric tensors

TL;DR

The paper develops a local, jacket-free approach to the 1/N expansion for tensor models with two symmetric rank-D tensors and a K_{D+1} interaction. By analyzing embeddings, ring structures, and, crucially, dipole deletions, it proves that all connected graphs have nonnegative degree and that degree-0 graphs are exactly melonic, even in the symmetric case. The method avoids global jacket constructions and relies on systematic local graph moves to show that degree does not increase under deletions, leading to dominance of melonic graphs at leading order. This establishes a melonic 1/N expansion for models with two symmetric tensors and highlights a robust route to universality in symmetric tensor theories.

Abstract

It is well known that tensor models for a tensor with no symmetry admit a $1/N$ expansion dominated by melonic graphs. This result relies crucially on identifying \emph{jackets} which are globally defined ribbon graphs embedded in the tensor graph. In contrast, no result of this kind has so far been established for symmetric tensors because global jackets do not exist. In this paper we introduce a new approach to the $1/N$ expansion in tensor models adapted to symmetric tensors. In particular we do not use any global structure like the jackets. We prove that, for any rank $D$, a tensor model with two symmetric tensors and interactions the complete graph $K_{D+1}$ admits a $1/N$ expansion dominated by melonic graphs.

Figures (13)

• Figure 1: The vertex and propagator of the model for $D=3$, and the vertex and some of the contributions to the propagator in $D=4$.
• Figure 2: The (melonic) ring graph and a more complicated melonic graph for $D=3$.
• Figure 3: A graph with no jackets in $D=3$.
• Figure 4: Ring graphs.
• Figure 5: The melonic graph with two vertices in $D=3$.

Theorems & Definitions (31)

• Definition 1
• Lemma 1
• proof
• Remark 1
• Definition 2
• Lemma 2
• proof
• Proposition 1
• proof
• Proposition 2