### Local Nets and Co-Sheaves

#### Posted by Urs Schreiber

Before getting back to Bruce’s message about BV-quantization of Chern-Simons theory below I need to post some musings and references on the relevance – or not – of codescent for those co-presheaves known as local nets of of quantum observables. I was led to reconsider this after being pointed to the relevance of the split property for wedges of such nets and from some discussion with Jacques Distler.

It is a curious fact that one of the axiomatizations of quantum field theory, AQFT is centered around certain *co-presheaves* (equivalently: pre-cosheaves, if you prefer), called *local nets*, defined on certain subsets of Minkowski (or globally hyperbolic Lorentzian) spaces, but that the (co-)sheaf-theoretic consequences of this are rarely every explored in the existing literature.

Except possibly for work of John Roberts, who may be thinking of the co-presheaf co-cohomology (= homotopy !?) of local nets. I am not sure. In the article mentioned in Roberts on nonabelian cohomology he is motivated by assigning cohomology groups to nets, but then passes to the grander question of general nonabelian cohomology. Personally, it took me a bit to live up to the full generality of this idea and Ezra Getzler’s wise comment, but now we are getting there with a full description of differential nonabelian cohomology.

What makes me interested in the question of to which extent local nets are co-sheaves is that I realized that differential nonabelian cocycles on Minkowski spaces (hence things living in generalized nonabelian sheaf cohomology) come with their endomorphism co-presheaf which happens to be a local net.

And not only that. This canonical co-presheaf associated to a differential nonabelian cocycle happens to satisfy the *time slice axiom*. This is not quite the same as a co-sheaf condition but goes in that direction. I had not tried to check if endomorphism co-presheaves of differential cocycles perhaps happen to actually satisfy codescent and hence are actually co-sheaves. But then I was pointed out to me that something called the *split property for wedges* (see section 2.2 in G. Lechner’s thesis On the construction of quantum field theories with factorizing S-matrices) may play a role in my construction.

By a lemma due to Michael Müger, this in turn implies the *additivity* property, which looks like it should be close to a co-sheaf property. I am not fully sure about this because one needs to be careful due to the fact that these local nets take values in von Neumann algebras, which leads to subtleties concerning completions. I may be missing the obvious. All help is appreciated!

I’ll end with a somewhat speculative but maybe suggestive observation:

The chiral deRham complex is a sheaf of vertex operator algebras on target space #. On the other hand, vertex operator algebras are expected to give rise to local nets (on parameter space, i.e on the worldvolume) by smearing vertex operators with test functions supported in causal subsets. It seems not unnatural therefore to speculate that the chiral deRham complex may be regarded as something which is a sheaf with respect to target space and a co-sheaf with respect to parameter space (worldvolume). Would seem to make sense. But I don’t know.

**P. S.**

While searching for more material on this, I came across the curious document Non-abelian quantum algebraic topology, I. Weirdly enough, this has also been pointed out to me back then when Ezra Getzler had pointed me to nonabelian sheaf cohomology, but I had completely forgotten about it. Some things maybe just need their time in sub-conciousness.

Anyway, as far as I can tell this document is a sketch of an idea that apparently arose from interaction of Ronnie Brown’s expertise on Nonabelian algebraic topology – itself a crucial ingredient in nonabelian cohomology – with researchers interested in identifying good mathematical formalism for describing modern quantum physics. The document mentions local nets as co-presheaves and mentions the van Kampen theorem – but does not seem to try to relate them.(?)

I am not sure I can see yet where this project of *nonabelian quantum algebraic topology is headed* precisely, the present version of the above document currently has a bit the appearance of the result of a preliminary brainstorming, but the basic impetus sure resonates with some of my feelings: we are in a period where the true formalism underlying quantum field theory could be identified by putting the existing puzzle pieces together in the right way, and nonabelian algebraic topology and cohomology will apparently play a pivotal role in that. In one way or other.

And one of the concrete aspects of this question clearly is: how exactly do Haag-Kastler nets fit into the story of nonabelian cohomology?

## Re: Local Nets and Co-Sheaves

Regarding the relation of additivity property of local nets with a cosheaf property, you may want to have a look at Section 4 of the paper

- K. Fredenahgen, R. Haag, Commun.Math.Phys. 108 (1987) 91-115

There it’s discussed how to obtain the “gluing” property from additivity in terms of local *-representations. The result is closely related to Roberts’s net cohomology.