## June 15, 2006

### 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 nonabelian 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.

Posted at June 15, 2006 6:28 PM UTC

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### Re: Roberts on Nonabelian Cohomology

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

Duskin, J.
$K\left(\pi ,\phantom{\rule{thinmathspace}{0ex}}n\right)$-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)

Posted by: David Roberts on June 16, 2006 5:08 AM | Permalink | Reply to this

### Re: Roberts on Nonabelian Cohomology

Duskin has a number of further papers on the subject of higher descent.

An important first place to look is this paper by his former student:

Glenn, Paul G. Realization of cohomology classes in arbitrary exact categories. J. Pure Appl. Algebra 25 (1982), no. 1, 33–105.

In this paper, Glenn shows that all (weak) n-groupoids have a simplicial nerve - not only strict ones. (The story for higher categories is less clear, but most current applications in physics appear to involve higher groupoids, not there categorical generalizations.)

Here is another paper worth reading:

Duskin, J. An outline of a theory of higher-dimensional descent. Actes du Colloque en l’Honneur du oixantième Anniversaire de René Lavendhomme (Louvain-la-Neuve, 1989). Bull. Soc. Math. Belg. Sér. A 41 (1989), no. 2, 249–277.

In this paper, Duskin works out the theory of descent (=nonabelian sheaf cohomology) in detail for the 2-truncated case, and promises a sequel, never written, which does the full story.

Posted by: Ezra Getzler on September 3, 2006 8:04 PM | Permalink | Reply to this

### Re: Roberts on Nonabelian Cohomology

It should be useful to direct people to the web site on `Higher dimensional group theory
which also gives lots of references on nonabelian methods in algebraic topology and homological algebra.

The basic premise is that by passing from groups to groupoids one has a notion of n-fold groupoid, whereas n-fold groups are just abelian groups for n>1. It is known that these n-fold groupoids are at least as rich as homotopy n-types, and that special cases can be computed precisely as algebraic models of homotopy types. Also there are some ideas of relations with higher order symmetry, at least as far as level 3.

In effect, group theory is seen as level 1 of a theory extending in all dimensions. There are rich relations with nonabelian cohomology, since these higher structures, or many special cases of them, can serve as coefficients in such theories. A classical example of such coefficients, due to Dedecker in the 1960s, is the notion of crossed module, which is equivalent to a 2-fold groupoid in which one structure is a group.

Posted by: Ronnie Brown on August 23, 2006 12:44 PM | Permalink | Reply to this

### Re: Roberts on Nonabelian Cohomology

It should also be useful to direct the interested readers to: “Non-Abelian Quantum Algebraic Topology.”, (a 10th version of a monograph in preparation), with the emphasis placed on theoretical physics, especially Quantum Gravity and the Fundamental Properies of Space-Time, including the Quantum Fundamental Group of (non-commutative) Space-Time.

http://www.ag.uiuc.edu/~fs401/Quantum%20Algebraic%20Topology.pdf

which includes a substantial section on Roberts’ approach to ‘AQFT’ –in fact,
Non-Abelian Algebraic Topology of Quantum Field Theory: a largely self-contained presentation with a list of relevant references that may be useful to the student interested in either Theoretical Physics or/and Non-Abelian Algebraic Topology and its application to Quantum, string, Penrose’ ,Hawking’, Connes’, etc… physical theories.

Posted by: I.C. Baianu on September 2, 2006 11:29 PM | Permalink | Reply to this
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