## August 6, 2013

### Maybe this time …

For many years, I tried keeping up with the LQG literature. Though it provided occasional fodder for blogging, it mostly was an exercise in frustration. Years ago, I gave up the effort. Still, occasionally, an LQG paper crosses my radar screen with claims interesting enough to cause me to suspend my better judgement.

One such paper, by Gomes et al, purports to be a significant breakthrough in the understanding of AdS/CFT. They claim to reproduce the conformal anomaly of a boundary CFT from some Loopy formulation (“Shape Dynamics”) of the bulk theory, thereby shedding light on the 1998 computation of Henningson and Skenderis who first reproduced the conformal anomaly from AdS/CFT (a more careful and thorough derivation can be found in a followup paper).

How could I resist?

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

(1)$T(x) \equiv g^{\mu\nu}(x){\langle T_{\mu\nu}(x)\rangle}_g$

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

(2)$T(x) = \begin{cases} - \frac{c}{6} {(\text{Euler})}_2 &d=2\\ 2\left( c\, {(\text{Weyl})}^2 - a\, {(\text{Euler})}_4 \right)&d=4 \end{cases}$

where2

\begin{aligned} {(\text{Euler})}_2 &= \frac{1}{4\pi} R\\ {(\text{Weyl})}^2 &= \frac{1}{32\pi^2} {\left(C_{\mu\nu\lambda\rho}\right)}^2 = \frac{1}{32\pi^2} \left(R_{\mu\nu\lambda\rho}^2 -2 R_{\mu\nu}^2 +\tfrac{1}{3} R^2\right)\\ {(\text{Euler})}_4 &= \frac{1}{32\pi^2} \left(R_{\mu\nu\lambda\rho}^2 -4 R_{\mu\nu}^2 + R^2\right) \end{aligned}

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.

1 For definiteness, we work in Euclidean signature, with a closed Riemannian $d$-manifold, $M$.

2 The Euler densities are normalized so that $\chi(M) = \int_M {(\text{Euler})}_{d} d\text{Vol}_g$

Posted by distler at August 6, 2013 10:59 AM

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### Re: Maybe this time …

Very good analysis.

Posted by: Tienzen (Jeh-Tween) Gong on August 7, 2013 12:38 PM | Permalink | Reply to this

### Re: Maybe this time …

Dear Jacques,
Shape Dynamics has nothing to do with LQG, except that one of the authors of the SD paper (Lee Smolin) used to work on LQG.
Best regards,
Aleksandar Mikovic

Posted by: Aleksandar Mikovic on August 8, 2013 8:42 AM | Permalink | Reply to this

### Shape Dynamics

Shape Dynamics has nothing to do with LQG, except that one of the authors of the SD paper (Lee Smolin) used to work on LQG.

Loops 2013 had an entire parallel session and a plenary talk devoted to Shape Dynamics. While it doesn’t have anything to do with Ashtekar variables (and, hence, in a narrow technical sense, isn’t LQG), it is worked on by the same people with exactly the same motivation:

• the hope that, by choosing the “right” set of canonical variables, they will finally succeed in constructing a sensible quantum theory, despite the failure of their previous attempts.
Posted by: Jacques Distler on August 8, 2013 9:49 AM | Permalink | PGP Sig | Reply to this

### Re: Shape Dynamics

Clearly there are many different ways to canonically quantize gravity. However, these quantizations may not be equivalent, so if one set of variables is problematic, that does not mean that some other set will also be difficult. Hence problems and difficulties of shape dynamics do not imply problems for LQG. However, it is true that canonical quantization of GR in general is not easy and it is plagued with difficulties; and in that sense I understand your frustration with LQG. That is the reason why many people (including myself) switched to path-integral quantization of GR, which for LQG takes the form of spin-foam models. In this area the progress has been bigger, and recently it has been shown that the classical limit of spin-foam models is the area-Regge action and that a categorical generalization of spin-foam models (spin-cube models) have the classical limit which is the Regge action, which gives GR for sufficiently fine triangulations (see my papers on the effective action for spin-foam and spin-cube models).

Posted by: Aleksandar Mikovic on August 9, 2013 6:14 AM | Permalink | Reply to this

### Re: Maybe this time …

I will add that this is not the first time that some of these points have been explained to at least one of the authors of the paper, with essentially the same result.

Posted by: Robert McNees on August 11, 2013 7:57 PM | Permalink | Reply to this

### Not the first time?

Surely, you’re not referring to the author who frequently, publicly pontificates on the intellectual status of AdS/CFT. Because that would be really embarrassing (although it would explain a lot).

Posted by: Jacques Distler on August 11, 2013 10:37 PM | Permalink | PGP Sig | Reply to this

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