### Oberwolfach CFT, Arrival Night

#### Posted by Urs Schreiber

After supper I went jogging along the Wolf river through the tiny village Oberwolfach, from where one can peer up the black forest mountains and see the illuminated MFO library shining through the fir trees, with no other light source except for a bright full moon on a starlit sky. That’s probably about as romantic as math can get.

On the eve of the CFT workshop starting tomorrow, I am struggling with understanding how…

… how the Araki-Haag-Kastler-Axioms should fit into the grand scheme of things – in particular, how they relate to Segal’s axioms, and their refinement to extended QFTs.

I expect that the *local* in “Haag-Kastler local QFT” is the *extended* in “Stolz-Teichner/Freed-Hopkins extended QFT”.

I also expect that the passage

extended Segal functorial QFT $\to$ Haag-Kastler algebraic QFT

is, for $n$-dimensional QFT, essentially the $n$-fold categorification of the functor

Schrödinger picture $\to$ Heisenberg picture

which is essentially induced by postcomposition with the equivalence

Hilbert spaces with cyclic vacuum vector $\simeq$ $C^*$-algebras with pure normal state

along the lines discussed in QFT of Charged n-Particle: Sheaves of Observables.

There are various indications for how this should work, but I clearly don’t get the full picture yet.

My Guru says: *When in doubt, formulate the arrow theory – then follow the Dao.*

I’ll try to do that, and I am sure all I need to do is to meditate over these mandalas a little further.

Starting with an extended Segal-like functorial QFT, what does “forming the algebra of observables”, really mean? It turns out that when one writes it out, a nice pattern emerges: QFT of Charged $n$-Particle: Algebra of Observables.

This gives a “2-monoid of observables”, which, for $n=1$ reproduces the ordinary Weyl algebras coming from canonical positions and momenta on target space, while for $n\gt 1$ one can see on heuristic grounds that it produces the corresponding algebra, now for loop space, double loop space, and so on.

By taking these loops, double loops, etc, to really be functorial images
$\gamma : \mathrm{par} \to \mathrm{tar}$
of a parameter space $\mathrm{par}$ which is not just a single loop, say, but something resolving this into a category of paths in the loop, etc, one finds, at least heuristically again, that this Weyl algebra on loop space decomposes into lots of *local algebras* living over all these intervals.

Then feed all this into the general arrow-theory for disk correlators, as described in D-Branes from Tin Cans, III: Homs of Homs to find what the Dao wants to assign to boundaries.

As always, I’d just have to follow where the formalism leads me here, but for this step I have not managed to do so yet.

This is what my thoughts were revolving around as I jogged by the bank of the river Wolf.

It does not seem implausible that on a boundary interval $I$ this procedure produces the chiral algebra $A(I)$ living there, arising essentially as the restriction of the above bulk algebras $B$ to the constant paths starting at that boundary, and that the canonical inclusion $A \to B$ appears at the level of morphisms, thus reproducing the subfactor inclusion/Q-system/Frobenius algebra of DHR endomorphisms that one would expect here, along the lines that I mentioned in Some Notes on Local QFT.

But as yet I fail to see this in detail.

I realize that one reason for this is that, while I think I know what the $n$-algebra of observables itself is like, arrow theoretically, I have no good understanding yet of what the $n$-categorical Heisenberg picture evolution functor would actually look like on morphisms. This, in turn, is at least partly due to the fact that I have not yet fully used, in the present context, all information available about the categorification of the equivalence $\mathrm{Hilb}_{\mathrm{cyc}} \simeq C^*_{\mathrm{stat}}$ of Hilbert spaces with cyclic states and $C^*$-algebras with pure normal states on them. This will involve the categorified Gelfand-Naimark theorem.

One problem with not getting confused in this context here is that algebras are making an appearance in various different guises. Not every algebra one encounters here plays the same kind of role, and that entities of different nature appear to us as just algebras makes some important structures hard to discern.

For instance, I am wondering: is the phenomenon that AQFT works so very well for 2-dimensional QFTs (where it is a very powerful tool), but has as yet found no real application to physically nontrivial QFTs in higher dimensions (as far as I am aware), just a simple consequence of the fact that people study the simpler examples first, or is there maybe a connection to the fact that the von Neumann algebras playing such a prominent role naturally live precisely in a 2-category, namely the 2-category whose morphisms are their bimodules?

While authors of standard AQFT literature rarely think of algebras as living in the 2-category of bimodules, this fact is actually implicitly of utmost importance for most of the crucial constructions revolving around DHR endomorphisms: morphisms of morphisms of algebras of observables appear all over the place, being addressed as “intertwiners”. These intertwiners are really morphisms of the left modules induced by the respective algebra morphisms, hence really 2-morphisms in bimodules of von Neumann algebras.

I am being thrown back and forth between believing that this is just a coincidence of low dimensions which I’d better ignore in order not to confuse myself, or if it is rather a crucial hint that I should take serious and think through to the end.

Maybe the next days will help crystallize answers to these puzzling issues.

By the way, concerning Segal’s axioms of QFT, I discovered this interesting paper:

Doug Pickrell
*$P(\phi)_2$ quantum field theories and Segal’s axioms*

math-ph/0702077

## Re: Oberwolfach CFT, Arrival Night

So the obvious question then:

Vector

Hilbert Space

Von Neumann Algebra

?????????