On the canonical c-function in 4-d field theories possessing supergravity duals

We study monotonicity and other properties of the canonical c-function (defined through the correlator of the energy–momentum tensor) in some holographic duals of 4-d quantum field theories. The canonical c-function and its derivatives are related to the 5d Greenʹs function of the dual supergravity theory. While positivity of the canonical c-function is obvious, we have not found a general proof of its monotonicity, even though c is monotonic in the few explicit examples we examine in this Letter.