A Note on Scalar Field Theory in AdS_3/CFT_2

TL;DR

This paper develops a formalism for scalar fields in AdS_3, deriving boundary conformal fields, Virasoro algebra structures, and boundary CFT properties, including central charges and ghost-like features.

Contribution

It introduces a novel boundary formalism for scalar fields in AdS_3, deriving boundary conformal transformations, Virasoro algebras, and CFT characteristics from bulk dynamics.

Findings

Boundary scalar fields transform as conformal fields with specific dimensions.

Two copies of the Virasoro algebra emerge with zero classical central charge.

Quantum effects introduce non-zero central charges depending on scalar mass and AdS radius.

Abstract

We consider a scalar field theory in AdS_{d+1}, and introduce a formalism on surfaces at equal values of the radial coordinate. In particular, we define the corresponding conjugate momentum. We compute the Noether currents for isometries in the bulk, and perform the asymptotic limit on the corresponding charges. We then introduce Poisson brackets at the border, and show that the asymptotic values of the bulk scalar field and the conjugate momentum transform as conformal fields of scaling dimensions \Delta_{-} and \Delta_{+}, respectively, where \Delta_{\pm} are the standard parameters giving the asymptotic behavior of the scalar field in AdS. Then we consider the case d=2, where we obtain two copies of the Virasoro algebra, with vanishing central charge at the classical level. An AdS_3/CFT_2 prescription, giving the commutators of the boundary CFT in terms of the Poisson brackets at the border, arises in a natural way. We find that the boundary CFT is similar to a generalized ghost system. We introduce two different ground states, and then compute the normal ordering constants and quantum central charges, which depend on the mass of the scalar field and the AdS radius. We discuss certain implications of the results.