There seems to have been some discussion of Cardy's "a-theorem" recently:
“It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, Sd, when suitably regularized, defines a c-function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly.” said Cardy. “It is shown that the existence of such a c-function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.”
I would appreciate it if someone could give a reasonbly concise explanation of what the theorem states and implies, at a level for someone with a superficial understanding of quantum field theory.
For example, what is the sphere in the above quote and how does the existence of the c-function lead to the main conclusion of the a-theorem which is purportedly:
"... a multitude of avenues in which quantum fields can be energetically excited (a) is always greater at high energies than at low energies. "
Edit: I think I found what I needed. Section 4.4 of David Tong's string theory notes gives a nice explanation of the trace anomaly and the "a" and "c" theorems. I assume the sphere they were talking about in the Nature article is just the $S^4$ of Euclidean compactified spacetime, in which case "a" is its Euler characteristic.