Non-Perturbative Defects in Tensor Models from Melonic Trees

TL;DR

This paper investigates non-perturbative defects in the Klebanov-Tarnopolsky tensor model, revealing solvable large N defect dynamics, conformal defect solutions, and RG flows, including exact results for line defects and their entropies.

Contribution

It introduces a novel large N limit for defects in tensor models, enabling non-perturbative analysis and exact solutions for defect one-point functions and RG flows.

Findings

Defect one-point functions are computed non-perturbatively using melonic tree diagrams.

Exact RG flow solutions between trivial and conformal line defects in 4−ε dimensions.

Line defect entropy is calculated and the g-theorem is verified.

Abstract

The Klebanov-Tarnopolsky tensor model is a quantum field theory for rank-three tensor scalar fields with certain quartic potential. The theory possesses an unusual large $N$ limit known as the melonic limit that is strongly coupled yet solvable, producing at large distance a rare example of non-perturbative non-supersymmetric conformal field theory that admits analytic solutions. We study the dynamics of defects in the tensor model defined by localized magnetic field couplings on a $p$-dimensional subspace in the $d$-dimensional spacetime. While we work with general $p$ and $d$, the physically interesting cases include line defects in $d=2,3$ and surface defects in $d=3$. By identifying a novel large $N$ limit that generalizes the melonic limit in the presence of defects, we prove that the defect one-point function of the scalar field only receives contributions from a subset of the Feynman diagrams in the shape of melonic trees. These diagrams can be resummed using a closed Schwinger-Dyson equation which enables us to determine non-perturbatively this defect one-point function. At large distance, the solutions we find describe nontrivial conformal defects and we discuss their defect renormalization group (RG) flows. In particular, for line defects, we solve the exact RG flow between the trivial and the conformal lines in $d=4-\epsilon$. We also compute the exact line defect entropy and verify the $g$-theorem. Furthermore we analyze the defect two-point function of the scalar field and its decomposition via the operator-product-expansion, providing explicit formulae for one-point functions of bilinear operators and the stress-energy tensor.