# What is the relation of the mass of vector field in bulk and the scaling dimension of current operator in CFT?

In AdS/CFT correspondence, we know that,

$$m^2=\Delta(\Delta-d)$$ where m is the mass of a scalar field and $\Delta$ is the scaling dimension of the dual operator in CFT. What about the relation of the mass of vector field in bulk and the scaling dimension of current operator in CFT?

• Hi and welcome to the Physics SE! The equations become much easier to read, search and edit when mathjax is used. I've fixed this question, but it'd be great if you could use it in your next posts. – stafusa Dec 28 '17 at 16:46

## 1 Answer

A vector field $A_{\mu}$ with spin-1 in the bulk has a dual spin-1 operator on the field theory, let's call it $J_{\mu}$.

If the vector field is massless, then $\Delta_{J}= d-1$ and $J_{\mu}$ is a conserved current.

In any other case, the current is not conserved.

The relation you wanted

$$m^2 = R^2(\Delta − 1)(\Delta − 3)$$

where $R$ is the AdS radius and $\Delta$ is the conformal dimension.

More generally in the context of the AdS/CFT; gauge symmetries in the gravity theory correspond to global symmetries in the CFT.

One relevant reference is

http://laces.web.cern.ch/Laces/LACES09/notes/dbranes/lezioniLosanna.pdf

and of course, there are many more.

Cheers!!!