AdS/CFT Correspondence and Symmetry Breaking

TL;DR

This paper explores how the AdS/CFT correspondence can be used to analyze symmetry-breaking vacua in gauge theories, clarifying the relation between scalar fields in AdS and operators in the boundary CFT, including subtleties related to operator dimensions.

Contribution

It provides a detailed method to compute expectation values of scalar operators in symmetry-breaking vacua using AdS perturbations and clarifies the duality for scalar fields with negative mass-squared.

Findings

Calculation method for scalar operator expectation values from AdS perturbations.

Clarification of scalar fields with negative mass-squared and their dual operators.

Relation between different operator dimensions via Legendre transform of generating functionals.

Abstract

We study, using the dual AdS description, the vacua of field theories where some of the gauge symmetry is broken by expectation values of scalar fields. In such vacua, operators built out of the scalar fields acquire expectation values, and we show how to calculate them from the behavior of perturbations to the AdS background near the boundary. Specific examples include the ${\cal N}=4$ SYM theory, and theories on D3 branes placed on orbifolds and conifolds. We also clarify some subtleties of the AdS/CFT correspondence that arise in this analysis. In particular, we explain how scalar fields in AdS space of sufficiently negative mass-squared can be associated with CFT operators of {\it two} possible dimensions. All dimensions are bounded from below by $(d-2)/2$; this is the unitarity bound for scalar operators in $d$-dimensional field theory. We further argue that the generating functional for correlators in the theory with one choice of operator dimension is a Legendre transform of the generating functional in the theory with the other choice.