Comments on Chemical Potentials in AdS/CFT

TL;DR

This paper introduces a method to identify holographic chemical potentials in AdS/CFT, ensuring consistency with thermodynamics and Legendre transformations, especially in cases with nontrivial charge distributions.

Contribution

It proposes a new approach to defining chemical potentials holographically, clarifying the role of gauge field shifts and charge distributions in the bulk geometry.

Findings

The method aligns chemical potential definitions with thermodynamic principles.

Constant gauge shifts are absent when Legendre transformations are well-defined.

Nontrivial charge distributions complicate the chemical potential definition.

Abstract

We propose a method for identifying holographic chemical potentials of conserved charges. The guiding principle is the consistency of the identification with the thermodynamic relations and the Legendre transformation. We consider the baryon-charge chemical potential as an example, and explain why the degree of freedom of the constant shift of the bulk U(1) gauge field is absent when the Legendre transformation is well-defined. The method proposed here suggests that the definition of the chemical potential may be more complicated compared with the case of localized charge if we have a nontrivial charge distribution along the radial direction of the bulk geometry.