Correlation functions of boundary field theory from bulk Green's functions and phases in the boundary theory

TL;DR

The paper develops a systematic framework to compute boundary correlation functions from bulk Green's functions for Dirichlet, Neumann, and mixed (Robin) boundary conditions, introducing a tuning parameter $h$ that interpolates between boundary phases. It provides explicit Green's functions and boundary kernels on flat planes and spheres, and extends the analysis to AdS spaces with boundary at infinity, revealing a universal logarithmic Neumann correlator in the massless AdS case and mass-dependent Dirichlet/Neumann kernels for massive fields; it also describes how the boundary operator dimensions and correlator structures evolve as the boundary location moves from infinity to the AdS center. A hierarchical structure of correlators is uncovered, built from simple kernels $K_0$ and $K_n$ with conformal interpretation and recurrence relations, linking boundary conformal data to bulk dynamics. The work also discusses phase transitions between Neumann and Dirichlet regimes, the role of mass as a regulator, and the possibility of dual 2D CFT descriptions for the Neumann phase, offering insights into holography and boundary QFTs in curved and AdS backgrounds.

Abstract

In the context of the bulk-boundary correspondence we study the correlation functions arising on a boundary for different types of boundary conditions. The most general condition is the mixed one interpolating between the Neumann and Dirichlet conditions. We obtain the general expressions for the correlators on a boundary in terms of Green's function in the bulk for the Dirichlet, Neumann and mixed boundary conditions and establish the relations between the correlation functions. As an instructive example we explicitly obtain the boundary correlators corresponding to the mixed condition on a plane boundary $R^d$ of a domain in flat space $R^{d+1}$. The phases of the boundary theory with correlators of the Neumann and Dirichlet types are determined. The boundary correlation functions on sphere $S^d$ are calculated for the Dirichlet and Neumann conditions in two important cases: when sphere is a boundary of a domain in flat space $R^{d+1}$ and when it is a boundary at infinity of Anti-De Sitter space $AdS_{d+1}$. For massless in the bulk theory the Neumann correlator on the boundary of AdS space is shown to have universal logarithmic behavior in all AdS spaces. In the massive case it is found to be finite at the coinciding points. We argue that the Neumann correlator may have a dual two-dimensional description. The structure of the correlators obtained, their conformal nature and some recurrent relations are analyzed. We identify the Dirichlet and Neumann phases living on the boundary of AdS space and discuss their evolution when the location of the boundary changes from infinity to the center of the AdS space.