The n-Category Café

As you know, work in physics from time to time leads to interesting conjectures in mathematics. Usually what happens is that first experimental physicists measure something which leaves everyone more or less puzzled. After a while theoretical physicists are beginning to understand what’s going on and are growing fond of some idea which they then regard as established textbook fact. Then mathematical physicists come along and try to find a precise statement of this idea. Ideally this results in a well defined conjecture. Then finally mathematicians try to prove this. Or roughly like this. It’s an academic food chain.

For this food chain to work, it’s important that the end product at every step is digestible by the species in the next layer.

Recently, there was a rather remarkable example where this didn’t work:

Many theoretical physicists are very fond of something called AdS/CFT duality. It’s an example, or a realization, of what has been called the holographic principle, which is an idea going back to t’Hooft. I had recently mentioned that in Extended QFT and Cohomology II: Sections, States, Twists and Holography and Why Theoretical Physics is Hard…. But if you want to learn more, you should better look at something like this Introduction to the AdS/CFT correspondence.

But beware: that’s clearly within the second layer of the food chain, not any higher yet.

In fact, it is quite hard to track down more than less precise statements of what this conjectural correspondence actually says in detail. Still, the excitement about it is comparatively large.

Now, a while ago, there was an attempt by a mathematical physicist to process this conjecture from the second to the third and forth layer of the food chain:

K.-H. Rehren
Algebraic Duality
hep-th/9905179.

But something strange happened: people in the second layer complained that this work did in fact not actually process the output which they produced, Maldacena’s AdS/CFT conjecture, but instead something more or less completely different!

See Jacques Distler’s entry on Rehren Duality.

Now, it so happened that yesterday at the conference which I am attending here, Hanno Gottschalk had given a talk based on Rehren’s work:

Hanno Gottschalk
AdS/CFT correspondence: lessons from constructive field theory

Abstract: Following ideas of Dütsch and Rehren, we give a proof of the equivalence of the two prescriptions of the conformal boundary field in AdS/CFT (“duality”). We use rigorous Euclidean path integrals to define functional delta functions of boundary values and restrictions to the conformal boundary. Both, in the interacting and non-interacting case, divergences arise that have not been observed so far. These divergences can, however, be removed without doing damage to reflection positivity and symmetry. Interactions with IR cut-off are taken into account in D=2 dimensional Euclidean AdS. We comment on some intriguing novel features of the IR-problem which comes from the fact that boundary sources in AdS/CFT lead to shifts in the bulk interaction that create an infinite amount of energy. This talk is based on joint work with Horst Thaler, math-ph/0611006.

As you know, I missed that talk. But I was told that after the talk a couple of string theorists at the conference raised those objections again: whatever this work is about, it is not really about what they call AdS/CFT duality.

Now, to some extent this is of course not a surprise: AdS/CFT duality is supposed to related a quantum field theory called $N=4$ super Yang-Mills to closed quantum string field theory. This means that in its full form, this is extremely far indeed from anything remotely tractable by precise mathematics.

So nobody should expect mathematical physicists to come up with an axiomatization of anything but a simplified toy model version of the conjecture.

But, and that’s important, this is not what the complaints are about: the complaint which is being voiced is that “Rehren duality” is not even roughly the same idea as string theorist’s AdS/CFT.

Whatever the reason for this disagreement (personally, I think there are problems with both layers of the food chain involved here) it is good that people are talking about it. Communication is important.

In our dicussion session, Matthias Blau played the role of the spokesman of the critics. His main emphasis was on this important fact:

The original AdS/CFT duality conjecture is not a statement which relates a quantum field theory in higher dimensions to whatever quantum field theory (or whatever) is obtained as the restriction of the former one to some boundary of its paremeter space.

It’s more indirect than that. The way I would phrase it is this:

The states of the higher dimensional QFT encode the correlators of a lower dimensional theory on the boundary.

So, you see, it’s not the states which restrict to states. But states restrict to correlators.

But, in saying so, I am already trying to move upwards in the food chain a little. What one finds in standard literature on AdS/CFT (like this Introduction to the AdS/CFT correspondence) looks like this:

On one side of the duality, we have something like a field theory (a string field theory, really, the second quantization of a certain 2-dimensional CFT) in higher dimensions, which is encoded by something like a path integral $$\int D\phi \exp(i S[\phi])$$ on a “exponentiated action” functional $\phi \mapsto \exp(i S[\phi])$ on some space of “fields” on some parameter space $X$.

On an (asymptotic) boundary $\partial X$ of parameter space $X$ one may fix the value of these fields $$\phi_0 := \phi|_{\partial X}$$ and then consider the path integral over the space of fields with that fixed restriction, only. This yields a functional $$\phi_0 \mapsto \exp(i W[\phi_0]) := \int_{\phi_{\partial X} = \phi_0} D\phi \exp(i S[\phi])$$ on fields on the boundary.

Now, naively one might think that this functional $\exp(i W[\cdot])$ should now be regarded as the action functional on fields $\phi_0$ on $\partial X$ and thus define a new field theory on $\partial X$.

But that’s at least not what the original AdS/CFT correspondence is about! Rather, this correspondence states that $\exp(i W[\cdot])$ is the generating functional of correlators of some field theory on the boundary.

This means that variations $$\frac{\delta}{\delta \phi_0(x_1)} \cdots \frac{\delta}{\delta \phi_0(x_n)} \exp(i W[\phi_0])|_{\phi_0 = 0} = \langle F(x_1) \cdots F(x_n) \rangle$$ are the correlator of some quantum field theory on the boundary. In terms of path integrals this reads $$\langle F(x_1) \cdots F(x_n) \rangle := \int D A\; F(x_1) \cdots F(x_n) \exp(i S_{\mathrm{bd}}[A]) \,.$$

I am glossing over mountains of details here, for good reason. The claim is that it is this broad structure

states in higher dimensions $\leftrightarrow$ correlators in lower dimensions

which is crucial to AdS/CFT – and which is not captured by those attempted axiomatizations by Rehren and others.

I think that instead of trying to tackle the full AdS/CFT correspondence, which is about theories in 10 and 4 dimensions (5 dimensions somehow disappear under some carpet, a non-rigorous carpet, of course), mathematical physicists should first try to understand the analogous situation in 3 and 2 dimensions.

This is the situation that Witten’s recent work applies to, which John very nicely summarized in TWF 254.

And for non-gravitational versions of this, there is already a pretty good mathematical understanding. I mention this from time to time here. Last time in FFRS on Uniqueness of Conformal Field Theory.