On two pieces of folklore in the AdS/CFT duality

TL;DR

This paper clarifies misconceptions in AdS/CFT duality, especially regarding symmetry correspondence and response functions, using holographic superconductors to interpret the holographic London equation.

Contribution

It provides a detailed analysis of the relationship between bulk and boundary symmetries and their impact on response functions in holographic models.

Findings

Global symmetry can become local when coupled with external sources.

The GKP-Witten relation yields response functions rather than Green functions.

Proper interpretation of the holographic London equation is essential.

Abstract

In the AdS/CFT duality, it is often said that a local symmetry in a bulk theory corresponds to a global symmetry in the corresponding boundary theory, but the global symmetry can become local when one couples with an external source. As a result, the GKP-Witten relation gives a response function instead of a Green function. We explore this point in details using the example of holographic superconductors. We point out that these points play a crucial role to interpret the holographic London equation properly.