Scalar Field Theory in the AdS/CFT Correspondence Revisited

TL;DR

This paper analyzes how different boundary conditions in scalar field theories within the AdS/CFT framework affect boundary two-point functions and conformal dimensions, revealing new normalizations and conditions for unitarity bounds.

Contribution

It provides a detailed study of Dirichlet, Neumann, and mixed boundary conditions, including their impact on two-point functions and conformal dimensions in AdS/CFT correspondence.

Findings

Dirichlet boundary condition yields no double zero at conformal dimension d/2.

Neumann boundary condition introduces new normalizations for two-point functions.

Mixed boundary conditions produce a family of conformal dimensions depending on parameters.

Abstract

We consider the role of boundary conditions in the $AdS_{d+1}/CFT_{d}$ correspondence for the scalar field theory. Also a careful analysis of some limiting cases is presented. We study three possible types of boundary conditions, Dirichlet, Neumann and mixed. We compute the two-point functions of the conformal operators on the boundary for each type of boundary condition. We show how particular choices of the mass require different treatments. In the Dirichlet case we find that there is no double zero in the two-point function of the operator with conformal dimension $\frac{d}{2}$. The Neumann case leads to new normalizations for the boundary two-point functions. In the massless case we show that the conformal dimension of the boundary conformal operator is precisely the unitarity bound for scalar operators. We find a one-parameter family of boundary conditions in the mixed case. There are again new normalizations for the boundary two-point functions. For a particular choice of the mixed boundary condition and with the mass squared in the range $-d^2/4<m^2<-d^2/4+1$ the boundary operator has conformal dimension comprised in the interval $[\frac{d-2}{2}, \frac{d}{2}]$. For mass squared $m^2>-d^2/4+1$ the same choice of mixed boundary condition leads to a boundary operator whose conformal dimension is the unitarity bound.