We describe a theory that
We start with a definition of differential nonabelian cohomology that works in great generality in any (∞,1)-topos whose underlying topos is a lined topos with an interval object that induces a notion of paths. In low categorical dimension this reproduces the description in [BaSc][ScWaI, ScWaIII]
We demonstrate that differential nonabelian cocycles are encoded by Ehresmann ∞-connections. Further assuming that the ambient (∞,1)-topos is a smooth (∞,1)-topos that admits a notion of ∞-Lie theory we show that these refine to Cartan-Ehresmann ∞-connections expressed in terms of ∞-Lie algebroid valued differential forms on the total space of a principal ∞-bundle. These are the structures studied in [SaScStI].
While it is well known that differential abelian cohomology models - and was largely motivated by the description of - abelian gauge fields in quantum field theory, many natural examples are in fact differential nonabelian cocycles. For instance the differential refinements of String structures and of Fivebrane structures or the field content of supergravity in the D'Auria-Fre formulation of supergravity.
In particular we exhibit classes of examples of cocycles in differential nonabelian cohomology arising from obstruction problems in twisted cohomology that capture the anomaly cancellation Green-Schwarz mechanisms in quantum field theory: we show that differential twisted cohomology captures phenomena such as twisted differential String- and Fivebrane structures studied in [SaScStII, SaScStIII ]
This is part of the project Differential Nonabelian Cohomology. See there for some background, history and references.
The following page gives an overview. The material itself is at the links in the table of contents.
and in that
need not be abelian or stable . It may in particular be an arbitrary homotopy n-type. More generally, may be an arbitrary ∞-groupoid. Possibly with extra structure, such as the smooth structure of a ∞-Lie groupoid.
This general notion is usually called nonabelian cohomology.
For ordinary abelian generalized (Eilenberg-Steenrod) cohomology there is a well known prescription for how to refine that to differential cohomology. Differential cohomology is to cohomology as fiber bundles are to bundles with connection. Differential nonabelian cohomology generalizes this to nonabelian cohomology.
The subject of differential nonabelian cohomology is the differential refinement of nonabelian cohomology – a refinement that is to abelian differential cohomology as nonabelian cohomology is to generalized (Eilenberg-Steenrod) cohomology.
We list the basic steps, constructions and theorems along which the theory proceeds. Links to pages that provide the technical details are provided.
In order to extract differential cohomology in the context given by some (∞,1)-topos we need to have a notion of parallel transport along paths in the objects of . This is encoded by assigning to each object its path ∞-groupoid . Morphisms enocode flat -valued parallel transport on or equivalently -valued local systems on .
Structurally, regarding the path ∞-groupoid assignment
In the presence of a notion of path ∞-groupoid we take flat differential -valued cohomology to be the cohomology with coefficients in an object in the image of this right adjoint, and write
A cocycle in flat differential -cohomology may be thought of as a local system with coefficients in .
There is a natural morphism
that includes each object as the collection of 0-dimensional paths into its path ∞-groupoid. This induces correspondingly a natural morphism of coefficient objects .
Lifting an -cocycle through this morphism to a flat differential -cocycle means equipping it with a flat connection.
We identify general non-flat differential nonabelian cocycles with the obstruction to the existence of the lift through from bare -cohomology to flat differential -cohomology.
There are two flavors of this obstruction theory whose applicability depeends on whether is group-like (once deloopable) or more generally nonabelian.
In the general case of non-group-like we shall use a [[nLab:Chern-character] morphism to approximate by an object that is group-like (and in fact fully abelian). We will demonstrate that this approximation leads in fact to an immediate generalization of the definition of abelian differential cohomology in terms of homotopy fibers of the Chern character map.
where is the coefficient for nonabelian deRham cohomology with coefficients in : a cocycle is a flat differential -cocycle whose underlying ordinary -cocycle is trivial.
This being a fibration sequence means that the obstruction to lifting an -cocycle to a flat differential -cocycle is the -cocycle given by the composite map . The class of this -cocycle we call the curvature characteristic class of the original -cocycle.
If this obstruction does not vanish but is given by some fixed curvature characteristic class , there is a general notion of twisted cohomology that encodes the -twisted-flat (namely: non-flat) differential cocycles.
This is finally our definition of differential nonabelian cohomology with grouplike coeffiecients.
For a grouplike object in and for a fixed curvature characteristic class, the differential -cohomology with curvature characteristic is the homotopy pullback
We show that the objects in correspond to diagrams in of the form
Here the interpretation of each layers is as indicated, as discussed in more detail after the next subsection.
If is not grouplike, one may use grouplike approximations to and proceed with these through the above constructions. One drastic but useful grouplike approximation to any is given by an integral Chern character morphism
We sow that from this one obtains along the above lines an object equipped with a morphism such that the induced morphism on cohomology
Proceeding with this morphism as a substitute for the non-existing produces a the notion of differential nonabelian cohomology that measures obstructions to flatness only up to some approximation, but that is direct generalization of the defintition of classical abelian differential cohomology in terms of homotopy fibers of the Chern character map.
Therefore we may in the following – for simplicity of notation but also suggestively – write for either the delooping of , if it exists, or ellse for the delooping of the codomain of the Chern character map of .
We show how the above differential cocycles constitute an -groupoidal version of the notion of Ehresmann connection.
This is achieved by noticing that every cocycle trivializes on the total space of the principal ∞-bundle that it classifies. On , we have the vertical path ∞-groupoid? of fiberwise paths. We show that every differential cocycle encoded by a diagram as above gives rise to a diagram
where each horizontal morphism is a cocycle in nonabelian deRham cohomology.
This we call a Cartan-Ehresmann ∞-connection. This finally constitutes concrete ∞-Lie algebroid valued differential forms data on a principal ∞-bundle. These -algebra connections have been discussed in
H. Sati, U. S., J. Stasheff, -algebra connections and their applications to String- and Chern-Simons -transport in B. Fauser et al. (eds.) Quantum Field Theory – Competetive Models, Birkhäuser (2009) (arXiv)
H. Sati, U. S., J. Stasheff, Twisted differential String- and Fivebrane structures, (arXiv)
This concludes the extended abstract . For details of the theory proceed with