The 1/N expansion of multi-orientable random tensor models

TL;DR

This work extends the $1/N$ expansion from colored tensor models to the broader class of multi-orientable (m.o.) tensor models by introducing the i.i.d. m.o. framework and developing a generalized jacket formalism. It proves that the leading contributions at large $N$ arise from melonic graphs, even though m.o. graphs form a strictly larger graph class, with non-bipartite graphs contributing subleading terms due to non-orientable jackets. The authors derive the $1/N$ scaling $A(\mathcal{G})=\lambda^{v_{\mathcal{G}}}N^{3-\varpi(\mathcal{G})}$ and show that $\varpi(\mathcal{G})=0$ defines the melonic sector, which maps to 3-ary trees and is countable via Catalan numbers. These results broaden the applicability of large-$N$ techniques in random tensor models and pave the way for future studies of m.o. tensor field theories, renormalization, and GFT generalizations.

Abstract

Multi-orientable group field theory (GFT) has been introduced in A. Tanasa, J. Phys. A 45 (2012) 165401, arXiv:1109.0694, as a quantum field theoretical simplification of GFT, which retains a larger class of tensor graphs than the colored one. In this paper we define the associated multi-orientable identically independent distributed multi-orientable tensor model and we derive its 1/N expansion. In order to obtain this result, a partial classification of general tensor graphs is performed and the combinatorial notion of jacket is extended to the multi-orientable graphs. We prove that the leading sector is given, as in the case of colored models, by the so-called melon graphs.

Figures (17)

• Figure 1: Propagator and vertex of the m.o. model
• Figure 2: Tensor graphs classification.
• Figure 3: Example of a graph with a tadface which is edge-colorable. The tadface line is shown in blue. The numbers $0$, $1$, $2$, $3$ are the color labels of the edges.
• Figure 4: On the left one can see the planar double tadpole as an example of a m.o. graph which is not colorable. On the right is pictured the "twisted sunshine" as an example of a m.o. graph which is 4-edge colorable but does not occur in colorable models.
• Figure 5: A 4-edge colorable m.o. graph which is not bipartite.

Theorems & Definitions (9)

• Proposition 3.1
• Proposition 3.2
• Definition 4.1
• Proposition 4.1
• Definition 4.2
• Theorem 6.1
• Proposition 6.1
• Proposition 6.2
• Theorem 6.2