Large N Field Theory and AdS Tachyons

TL;DR

This work clarifies how double-trace deformations in large-$N$ 4d field theories control conformal invariance and connect to AdS/CFT via tachyonic bulk modes. The authors prove that the double-trace beta function is quadratic in the coupling to all orders in planar perturbation theory, and derive the exact expressions for the running, the operator dimension, and the effective potential in terms of discriminant data $D(\lambda)$. They show that real fixed points preserve conformality, while negative discriminant leads to a Coleman–Weinberg instability and dynamical symmetry breaking, with the boundary result encoded in a complex scaling dimension ${\Delta}_{\cal O}=2\pm i b$. The AdS interpretation equates $m^2(\lambda)R^2$ to $-4+D(\lambda)$, relating bulk tachyon masses to boundary conformality and symmetry breaking, and is illustrated by explicit orbifold examples. Overall, the paper provides a concrete, testable framework to study conformality and instabilities in large-$N$ non-supersymmetric theories and their gravity duals.

Abstract

In non-supersymmetric orbifolds of N =4 super Yang-Mills, conformal invariance is broken by the logarithmic running of double-trace operators -- a leading effect at large N. A tachyonic instability in AdS_5 has been proposed as the bulk dual of double-trace running. In this paper we make this correspondence more precise. By standard field theory methods, we show that the double-trace beta function is quadratic in the coupling, to all orders in planar perturbation theory. Tuning the double-trace coupling to its (complex) fixed point, we find conformal dimensions of the form 2 + i b, as formally expected for operators dual to bulk scalars that violate the stability bound. We also show that conformal invariance is broken in perturbation theory if and only if dynamical symmetry breaking occurs. Our analysis is applicable to a general large N field theory with vanishing single-trace beta functions.

Figures (6)

• Figure 1: Proposal for the qualitative behavior of a "tachyon" mass in a freely acting orbifold, as a function of the 't Hooft coupling $\lambda$. The field is an actual tachyon (violating the BF stability bound) for $\lambda < \lambda_C$. See section \ref{['examples']} for more comments.
• Figure 2: One-loop contributions to the effective action from a diagram with two quartic vertices. Each vertex contributes a factor of $\lambda N$ and each propagator a factor of $1/N$, as indicated in (a). There are two ways to contract color indices: a single-trace structure (b), or a double-trace structure (c).
• Figure 3: Sample diagrams contributing to $\beta_f$ at one loop: (a) $v^{(1)}f^2 \;$; (b) $2 \gamma^{(1)}\lambda f \;$; (c) $a^{(1)}\lambda^2$.
• Figure 4: Feynman rules for (\ref{['Ltree']}).
• Figure 5: Diagram (a) is leading at large $N$, of order $O(1)$, but it is reducible. Diagram (b) is irreducible but it is subleading at large $N$, of order $O(1/N^2)$.