### The Manifold Geometries of QFT, I

#### Posted by Urs Schreiber

Spent today over at the Max-Planck Institute for Math in Bonn, close by the Hausdorff Institute, attending the first day of the conference

The manifold geometries of quantum field theory.

Here are some notes on what I have heard, concerning a) perturbative AQFT, b) algebraic AdS/CFT and c) rigorous path integrals for Chern-Simons theory.

**a) perturbative AQFT**

Klaus Fredenhagen and collaborators have in the course of several years been thinking about renormalization and renormalization group flow in the context of algebraic quantum field theory. Older work is for instance

Michael Dütsch, Klaus Fredenhagen
*Perturbative Algebraic Field Theory, and Deformation Quantization*

arXiv:hep-th/0101079

but the latest developments are as yet unpublished. Therefore I was eager to take notes in the talk. I was doing comparatively fine until about halfway through the talk. But right when it got to the main points I gave up, exhausted. The result as far as I got is here.

**b) algebraic AdS/CFT**

You may remember from the discussion here at the Café and in particular over at Jacques Distler’s blog here and more recently here that K.-H. Rehren made a proposal for how to phrase the AdS/CFT duality conjecture in a rigorous context. But there were complaints that this formalization didn’t capture the intended non-rigorous conjecture.

Roughly, the duality conjecture relates fields in an AdS space with *sources* for fields in a local field theory on the asymptotic boundary.

On the other hand, K.-H. Rehren observed that from the geometry of AdS space it follows directly that certain nets of observables on the asymptotic boundary can be regarded as the direct restriction of certain nets of observables on the full AdS space and proposed that this is what is going on in the original conjecture.

Aware that this direct restriction which sends fields in the bulk to their restriction on the boundary does not seem to harmonize with the original conjecture, he argued with Dütsch in arXiv:hep-th/0204123 that despite this appearance, both prescriptions do agree after all and that they are related by a functional Fourier transformations.

In discussion, Jacques Distler pointed out that for fields of positive mass one of the two sides of this would-be Fourier transform do not make sense (it’s actually a rather elementary observation in this context).

Now, today I heard H. Thaler talk about this. He has worked with H. Gottschalk on these functional Fourier transformations involved here, making sense of them beyond the perturbative expansion. I haven’t had a chance yet to take a closer look at their article

H. Gottschalk, H. Thaler
*AdS/CFT correspondence in the Euclidean context*

arXiv:math-ph/0611006

but am being told that their result can be summarized like this:

the transformation considered by Dütsch and Rehren makes sense (only) for a certain region of *negative* mass-squared $m^2$ of the scalar field under consideration, i.e. for tachyonic fields, namely for the case that the expression $\Delta_\pm := \frac{d}{2} \pm \frac{1}{2}\sqrt{d^2 + 4 m^2}$ ($d$ the number of dimensions) is positive for both signs of the square root. Sure enough, this is precisely what Jacques Distler pointed out, of course.

In the allowed tachyonic cases they can compute both sides of the transformation rigorously – and find that they vanish identically. All this seems to confirm that indeed Rehren’s “algebraic holography” proposal does not capture the idea of the original AdS/CFT duality conjecture.

**c) rigorous path integrals for Chern-Simons**

Last talk today was by A. Hahn, who has been working on bridging the gap between the non-rigorous but powerful path-integral description of Chern-Simons theory and the rigorous but less immediate algebraic description along the lines of Turaev and others.

There are a couple of algebraic invariants which have been constructed and which more or less suggest which CS-path integral quantities they should correspond to. For some this has now been realized. A. Hahn was in particular talking about Turaev’s *shadow invariants*. These are invariants on 3-manifolds which are products of the form $\Sigma \times S^1$ obtained by taking Wilson line observables and projecting them into the $\Sigma$-factor (hence looking just at their “shadow” on $\Sigma$.)

A. Hahn identifies the candidate path integral observable, chooses a convenient gauge fixing and then makes use of the fact that after the gauge fixing the path integral measure becomes Gaussian. The class of Gaussian path integral measures can be made sense of usining stochastic theory, white noise, etc. So that’s what he does. And he proves that the well-defined invariant he gets out of the path integral this way is indeed Turaev’s *shadow* invariant.

Details are in

Sebastian de Haro, Atle Hahn
*The Chern-Simons path integral and the quantum Racah formula*

arXiv:math-ph/0611084

## Re: The Manifold Geometries of QFT, I

The link to the Fredenhagen notes is broken.