Holography for degenerate boundaries

TL;DR

The paper extends the AdS/CFT correspondence to negatively curved Einstein manifolds with degenerate conformal boundaries by classifying two main boundary-types, analyzing the breakdown of local counterterm regularisation at finite radius, and proposing a finite-radius holographic dictionary $I_{\rm bulk}(\epsilon) \approx W_{\rm cft}(\epsilon)$. It provides concrete checks on Bergman-type geometries and on $H^{2} \times H^{2}$, showing that bulk divergences at finite $\epsilon$ match regulated boundary partition functions and that a bulk scalar can source a spectrum of boundary operators with varying conformal weights. The work demonstrates how correlation functions can be computed in these settings, revealing that reduced boundary symmetry allows for mixing of operators with different conformal weights and a partially constrained two-point structure. Collectively, these results generalize holography to degenerate boundaries, clarify the IR/UV correspondence in this context, and offer a framework for analyzing supergravity in backgrounds like $AdS_{3} \times H^{2}$, with explicit examples and scalar-field analyses supporting the construction.

Abstract

We discuss the AdS/CFT correspondence for negative curvature Einstein manifolds whose conformal boundary is degenerate in the sense that it is of codimension greater than one. In such manifolds, hypersurfaces of constant radius do not blow up uniformly as one increases the radius; examples include products of hyperbolic spaces and the Bergman metric. We find that there is a well-defined correspondence between the IR regulated bulk theory and conformal field theory defined in a background whose degenerate geometry is regulated by the same parameter. We are hence able to make sense of supergravity in backgrounds such as $AdS_{3} \times H^{2}$.