Explanation of Cardy's “a theorem”

Question

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.

Answer 1

Very loosely, what Cardy's $a$ theorem says is that when you measure a system at low energies you can see fewer degrees of freedom than when you measure it at high energies. For example if I take a salt crystal and hit with a high energy gamma ray I can knock at a chlorine nucleus, sodium nucleus, an inner shell electron an outer shell electron, I can excite spins. All sorts of degrees of freedom. But if I hit the salt crystal with something really low energy, like a slow moving neutron or something, all I can excite is a low energy phonon. So low energy means less degrees of freedom.

Now this should seem a little obvious. Of course if I have less energy I can do less. On the other hand the things that happen at high energy in our salt crystal (a free flying nucleus) look a lot different from the things that happen at low energy (a phonon, which is a collective of all the ions in the material). So its not obvious how to count "degrees of freedom" to make our intuition precise. And there are plenty of examples where the low energy degrees of freedom look nothing like the high energy degrees. A type II superconductor at high energy has boring electrons but at low energies has the motion magnetic flux vortices.

So in some sense the question is how to count degrees of freedom. Apparently a lot of the obvious guesses in (3+1)d are wrong. And the prescription provided by the $a$-theorem is kind of weird. For example a free gauge field (like E&M) has exactly 62 times as much "degrees of freedom" as a free scalar (like the Higgs boson). Not exactly what you would guess! But apparently if you try to count with that 62 changed to 61 or 63 you can find examples where there are more low energy "degrees of freedom" than at high energy. And now its proven.