Prismatic Large $N$ Models for Bosonic Tensors

TL;DR

This work introduces a stable, solvable bosonic tensor theory with $O(N)^3$ symmetry and a prism-like interaction (the prism term) that dominates its large-$N$ dynamics. The authors solve the model via Schwinger-Dyson equations and develop a $3- epsilon$ renormalized perturbation framework that includes eight $O(N)^3$-invariant sextic operators, revealing a real prismatic fixed point for sufficiently large $N$ and matching the SD results in the large-$N$ limit. They map out the bilinear operator spectrum across dimensions, identifying ranges free of complex dimensions ($2.81<d<3$ and $d<1.68$) and detailing the behavior of type A/C versus type B bilinears, including higher-spin towers and their large-spin limits. The analysis also yields $1/N$ corrections to operator dimensions and discusses the $d=1$ quantum-mechanical limit, where a discrete spectrum persists and signals possible AdS$_2$ interpretations, while offering a path to generalize to higher-rank tensors and supersymmetric extensions.

Abstract

We study the $O(N)^3$ symmetric quantum field theory of a bosonic tensor $φ^{abc}$ with sextic interactions. Its large $N$ limit is dominated by a positive-definite operator, whose index structure has the topology of a prism. We present a large $N$ solution of the model using Schwinger-Dyson equations to sum the leading diagrams, finding that for $2.81 < d < 3$ and for $d<1.68$ the spectrum of bilinear operators has no complex scaling dimensions. We also develop perturbation theory in $3-ε$ dimensions including eight $O(N)^3$ invariant operators necessary for the renormalizability. For sufficiently large $N$, we find a "prismatic" fixed point of the renormalization group, where all eight coupling constants are real. The large $N$ limit of the resulting $ε$ expansions of various operator dimensions agrees with the Schwinger-Dyson equations. Furthermore, the $ε$ expansion allows us to calculate the $1/N$ corrections to operator dimensions. The prismatic fixed point in $3-ε$ dimensions survives down to $N\approx 53.65$, where it merges with another fixed point and becomes complex. We also discuss the $d=1$ model where our approach gives a slightly negative scaling dimension for $φ$, while the spectrum of bilinear operators is free of complex dimensions.

Figures (13)

• Figure 1: Diagrammatic representation of the eight possible $O(N)^3$ invariant sextic interaction terms.
• Figure 2: Diagramatic representation of the Schwinger-Dyson equations. Solid lines denote $\phi$ propagators, and dashed lines denote $\chi$ propagators.
• Figure 3: Solving \ref{['eigenvaluenew']} for $d=2.9$.
• Figure 4: Solving \ref{['eigenvaluenew']} for $d=1$.
• Figure 5: The integration kernel for type B bilinears.