The 1/N expansion of colored tensor models in arbitrary dimension

TL;DR

The paper extends the 1/N expansion to colored group field theories in arbitrary dimension D, proving that leading-order graphs are dual to the sphere $S^D$ and that the free energy scales with $\delta^N(e)^{D-1}$. It introduces the framework of jackets and bubbles, establishes a jacket-based amplitude bound, and shows that planarity of all jackets selects the sphere topology as the dominant contribution. The authors provide a rigorous two-step induction, using 1-Dipole contractions to reduce graphs to a single D-bubble and ultimately identify the leading topology with $S^D$, with a special note on the D=3 case allowing a full topological expansion. These results offer analytic evidence for smooth, high-dimensional topologies dominating quantum gravitational sums in colored GFT models, including the $D=4$ Ooguri model.

Abstract

In this paper we extend the 1/N expansion introduced in [1] to group field theories in arbitrary dimension and prove that only graphs corresponding to spheres S^D contribute to the leading order in the large N limit.

Figures (3)

• Figure 1: Line and vertex of the Colored GFT graphs.
• Figure 2: The two point graph ${\cal G}_1$.
• Figure 3: Contraction of a 1-Dipole.

Theorems & Definitions (3)

• Definition 1
• Lemma 1
• Theorem 1