AdS/CFT Correspondence - Source Fields

Source Fields

Basically, the correspondence runs as follows; if we deform the CFT by certain source fields by adding the source

this will be dual to an AdS theory with a bulk field J with the boundary condition

where Δ is the conformal dimension of the local operator and k is the number of covariant indices of minus the number of contravariant indices. Only gauge-invariant operators are allowed.

Here, we have a dual source field for every gauge-invariant local operator we have.

Using generating functionals, the relation is expressed as

The left hand side is the vacuum expectation value of the time-ordered exponential of the operators over the conformal field theory. The right hand side is the quantum gravity generating functional with the given conformal boundary condition. The right hand side is evaluated by finding the classical solutions to the effective action subject to the given boundary conditions.