### Roberts on Nonabelian Cohomology

#### Posted by Urs Schreiber

I was scolded for never having cited

John E. Roberts
*Mathematical Aspects of Local Cohomology*

talk at Colloqium on Operator Algebras and their Applications to Mathematical Physics,

Marseille 20-24 June, 1977 .

Igor Baković was so kind to dig the paper out for me.

John E. Roberts ($\to $) is a leading figure of algebraic quantum field theory ($\to $) with many seminal results, among them the famous *Doplicher-Roberts theorem* ($\to $).

Algebraic quantum field theory (AQFT) is an attempt to guess a good set of axioms for quantum field theory, in the hope that proceeding rigorously from these axioms sheds light on the intricacies of QFT.

One of the main assumptions is that the local observables for quantum field theories on Minkowski spacetime should form a *local net of observable algebras*. Mainly, the idea is

1) to assign to each double lightcone an algebra of observables,

2) such that algebras of subcones embed into those of the lightcones containing them

3) and - most importantly - such that the algebras of spacelike seperated double cones mutually commute.

This is supposed to encode the physical concept of *microcausality*.

For more details see for instance the recent review

Hans Halvorson, Michael Müger,
*Algebraic Quantum Field Theory*

math-ph/0602036,

and in particular section 2 therein.

So Roberts, too, was interested in local nets of operators. In his paper cited above, he points out that interesting information about these can be obtained from *local cohomology* with values in some abelian group encoding information about these algebras.

Local cohomology with values in some local net of groups is pretty much like sheaf cohomology with values in a sheaf of groups.

In his paper, Roberts briefly notes that the second local cohomology associated to the net of solutions of the free vacuum Maxwell equations and of the free vector particle provides nontrivial information about the sheaf of Cauchy data for these fields.

After giving this example of an application to physics, however, the paper aims at a much more general goal, namely that of understanding what it could mean to have higher cohomology valued in *non*abelian groups.

Roberts was, apparently, the first one to notice that the only reasonable way this can be made sense of is in terms of labelling $p$-simplicies with $p$-morphisms of $n$-categories.

I am guilty of having studied an instance of this very general idea, without citing Roberts’ paper, for the case where the $n$-category in question is the 2-category of transport 2-funcors ($\to $) which locally describe 2-bundles with integrable connection.

You can find the tetrahedra decorated in pseudonatural transformations of transport 2-functors depicted here.

Of course, the point here is that one does not just postulate this structure of cohomology with values in categories, but one derives it from locally trivializing globally defined objects.

So, in fact, 2nd cohomology with values in 2-functor categories appears also when describing topological strings as well as structures appearing in 2D CFT ($\to $).

I would like to understand if there is any direct relation between the description of CFT appearing here ($\to $) and the way CFT as studied in the context of AQFT, as for instance in

K.-H. Rehren
*On local boundary CFT and non-local CFT on the boundary*

math-ph/0412049.

It *seems* to me that the axioms of AQFT are too strict to accomodate for full CFT on arbitrary (Euclidean) worldsheets, possibly with boundaries.

Certainly the assumption of Minkowski background structure is not applicable here, and the entire concept of local observables becomes rather ill-suited. But maybe that can be remedied?

Also, boundary conditions are crucial in 2D CFT. Essentially, one knows the full theory from just knowing everything about its boundary conditions (at least in the rational case). Attempts to formulate AQFT on spaces with boundaries have only rather recently appeared. So it might be too early to try to see the connection.

## Re: Roberts on Nonabelian Cohomology

Certainly people have been thinking about higher structures for ages. I’m surprised no one has cited

Duskin, J.

$K(\pi ,\phantom{\rule{thinmathspace}{0ex}}n)$-torsors and the interpretation of “triple”cohomology. Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2554–2557.

yet for discussing special cases of $n$-torsors, namely those associated to the “traditional” (i.e. abelian) higher cohomology groups. He works in the simplical category - really nerves of (strict) $n$-categories.

DM Roberts (no relation)