Scalar Field Theory in the AdS/CFT Correspondence Revisited

TL;DR

This work analyzes how Dirichlet, Neumann, and mixed boundary conditions for a bulk scalar in AdS/CFT alter the dual boundary CFT data. By constructing the appropriate boundary terms in the action and evaluating the on-shell action, the authors extract explicit boundary two-point functions and deduce the conformal dimensions Δ of the corresponding boundary operators, with careful treatment of special masses where logarithmic behavior appears. Key findings include: Dirichlet yields Δ = d/2 + ν (ν ≠ 0) or Δ = d/2 (ν = 0) or Δ = d (ν = d/2); Neumann yields Δ = d/2 + ν with notable cases Δ = (d-2)/2 at m^2 = 0 and Δ = d/2 at m^2 = -d^2/4; Mixed boundary conditions introduce a one-parameter family that, depending on ν and a parameter β, can give Δ in ranges such as Δ ∈ {d/2 − ν, (d−2)/2, d/2 + ν}, with possible transitions to the unitarity bound. Overall, the paper demonstrates that boundary terms and BC choices generically produce different boundary CFT correlators and operator dimensions, including special mass regimes and logarithmic structures, and discusses implications for higher-point functions and potential string-theoretic interpretations.

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.