Musings

The central object of study is the trace of the 1-point function of the stress tensor on a curved background¹

Naïvely, this vanishes in a CFT. In fact, when the spacetime dimension, $d$, is odd, it does vanish. But for $d$ even, there is an anomaly

where²

The trace anomaly coefficients ($c$ in $d=2$, $(a,c)$ in $d=4$) are constants which characterize, in part, the conformal field theory.

Note two obvious features

1. $T(x)d\text{Vol}_g$ is invariant under constant rescalings of $g$. As a consequence, the dependence of the effective action, $W[g]$, on the overall volume of $M$ is logarithmic, where the coefficient of the logarithm is proportional to the trace anomaly coefficient(s).
2. The group of Weyl transformations has a subgroup which preserves the overall volume of $M$ (the authors call these VPCTs). These act nontrivially, and integrating up that action yields a nontrivial dependence of $W[g]$ on the conformal factor (in $d=2$, this is the famous Liouville action).

Unfortunately, the authors got this backwards.

• They demanded invariance of $W[g]$ under VPCTs. This lead them to postulate Axiom 5’ of their paper, which states (in the notation of (1)) that $T(x)$ is a position-independent constant, for any choice of $g$. While that’s true for $d$ odd (where $T(x)=0$), it’s manifestly untrue for $d$ even (where it’s given by (2)).
• Conversely, they were labouring under the mistaken impression that $W[g]$ has some complicated dependence on the overall volume of $M$, which can — for large volume — be expanded in an asymptotic expansion. And it is this latter dependence that they proceed to compute from their loopy considerations.

In other words, their ‘significant breakthrough’ was to get the entire story precisely backwards.

Aughh!

After picking myself up off the field, and brushing off the dirt, I emailed the authors. After a lengthier-than-necessary back-and-forth, they finally agreed that there was an issue, but assured me that they knew how to resolve it. Weeks passed. Eventually, they sent me a revised version. In it, they withdrew the central claim, namely that they could reproduce the correct expression for (1) from some loopy bulk computation. Instead, they admitted that their loopy prescription computes a quantity entirely unrelated to the stress tensor of the boundary theory but claimed (incorrectly) that, in the large volume limit, both quantities vanish — so they’re morally the same.

An even lengthier discussion ensued, which slowly devolved into a tutorial on the most elementary aspects of AdS/CFT. They had no clue, for instance, how the metric $g$, of the boundary theory, is supposed to be related to the metric $G$, of the bulk theory (the most basic part of the AdS/CFT correspondence). Rather than picking up any of the well-written introductions to AdS/CFT, they apparently had simply decided to guess how the correspondence is supposed to work.

It seems to me that, if you

1. Guess what the AdS/CFT dictionary is supposed to be.
2. Use your guess to compute the first obvious observable (the trace anomaly of the boundary theory).
3. Obtain the wrong answer for that observable.

you might wish to re-examine your guess, instead of writing a paper claiming it to be a great conceptual breakthrough.

But that’s the kind of naïveté that keeps getting me (and Charlie Brown) in trouble.