Schreiber
Differential Nonabelian Cohomology

Differential nonabelian cohomology studies the generalization of the notion of connection on a bundle, similarly the notion of twisted local systems, to the context of higher geometry.

Parts of this generalization is known as differential cohomology refinements of abelian generalized (Eilenberg-Steenrod) cohomology theories by differential form data on smooth manifolds; such as Deligne cohomology and differential K-theory.

More generally, in differential nonabelian cohomology

• the cohomology being differentially refined may be nonabelian cohomology, classifying gerbes and, generally, principal ∞-bundles and ∞-vector bundles;

• the base spaces may be objects in higher geometry, known as orbifolds and, generally, ∞-stacks;

• the smooth spaces may be general ones as encoded by synthetic differential geometry; including among smooth manifolds also for instance smooth loop spaces and smooth classifying spaces.

The archetypical application of abelian differential cohomology has been the description of higher abelian gauge fields in quantum field theory and string theory, such as the electromagnetic field in terms of line bundles with connection, and its generalization to the Kalb-Ramond field in terms of bundle gerbes with connection and the RR-fields in terms of differential K-theory.

The archetypical application for differential nonabelian cohomology are higher nonabelian gauge fields generalizing the Yang-Mills field and the field of gravity in terms of connections on principal bundles: they generalize to differential string- and fivebrane-structures in terms of Ehresmann ∞-connections on principal ∞-bundles.

Invitation

A connection on a bundle over a space is a structure that assigns to every smooth path in the space a morphism between the fibers of the bundle over the endpoints of the path, such that this assignment respects composition of paths: this is called parallel transport.

There are two main aspects to this: global and local nontriviality; cocycles and their differential refinement.

The bundle itself encodes to which extent the assignment is globally nontrivial. It is defined by a map from the space to some classifying object that describes how the bundle is glued from trivial bundles: such a map is a cocycle in a suitable cohomology theory. If the bundle is a line bundle, this is an example of an abelian cohomology theory, also known as generalized (Eilenberg-Steenrod) cohomology. But despite its name, this is just a special case of the general notion of cohomology. If the bundle is a $G$-principal bundle for a nonabelian group $G$, its classifying cocycle is in nonabelian cohomology. If the bundle is a vector bundle, then it depends: if it is regarded as an associated bundle to a principal bundle it is classified in nonabelian cohomology. But if it is regarded as a representative cocycle in K-theory, then it is in abelian cohomology. Here we look at the general notion of cohomology that encompasses all these cases. Only for emphasis of its generality we often say nonabelian cohomology .

On the other hand, the connection makes these structures locally nontrivial : on local regions where the bundle is trivial, the assignment of morphisms to paths is fixed by the assignment to infinitesimal paths. Such an assignment is a set of differential forms. The failure of this being trivial is the curvature of the bundle, captured by its Chern character, which is a cocycle in deRham cohomology. The differential cocycle encoded by the connection on the bundle is hence the structure that connects the cocycle classifying the underlying bundle with its Chern-character; the bundle with the obstruction to equipping it with a trivial identification of its fibers along paths.

A differential cocycle is slightly different in nature from the nonabelian cocycle and the Chern-character deRham-cocycle that it consists of. Its conceptual home is not cohomology but twisted cohomology. Specifically: a differential cocycle is a flat cocycle – a local system – twisted by a curvature characteristic class .

We can formalize this a bit more. Write $A$ for the classifying object, such that a morphism $X\to A$ is a cocycle classifying a bundle. Then the Chern character is a morphism $A\to A_{\mathrm{ch}}$, where $A_{\mathrm{ch}}$ classifies certain cocycles in deRham cohomology that arise as curvature characteristic classes of $A$-cocycles. The homotopy fiber of this morphism is $A_{\mathrm{flat}}$, the classifying object for $A$-cocycles with trivializable curvature classes.

Write $H(X,A)$ for the collection of $A$-cocycles, $A$-coboundaries etc. and write $H(X,A)$ for the corresponding cohomology classes. The fibration sequence $A_{\mathrm{flat}}\to A\to A_{\mathrm{ch}}$ induces an exact sequence in cohomology $H(X,A_{\mathrm{flat}})\to H(X,A)\to H(X,A_{\mathrm{ch}})$, that identifies flat differential cocycles as objects in the homotopy fiber of $H(X,A)\to H(X,A_{\mathrm{ch}})$ over the trivial cocycle $0:*\to H(X,A_{\mathrm{ch}})$.

From this we deduce the definition of non-flat differetial cohomology:

Differential $A$-cohomology with curvature class $P$ is the homotopy fiber of the nonabelian Chern character $H(X,A)\to H(X,A_{\mathrm{ch}})$ over the nonabelian deRham cocycle $P:*\to H(X,A_{\mathrm{ch}})$.

One checks that in situations where a familiar notion of Chern character is available this abstract definition does reproduce the familar definitions of connections on bundles and differential refinements. In fact, for $A$ a spectrum, this definition is often taken as the definition of abelian differential cohomology. This is described notably in

• Mike Hopkins, I. Singer, Quadratic functions in geometry, topology and physics (arXiv)

• Dan Freed, Dirac Charge Quantization and Generalized Differential Cohomology (arXiv)

But more generally, the coefficient object $A$ and the domain object $X$ should be allowed to be any generalized smooth space: any smooth ∞-stack.

In this general case, the crucial question is therefore: what is the morphism $\mathrm{ch}:A\to A_{\mathrm{ch}}$?

Or more in detail:

• What is nonabelian deRham cohomology – deRham cohomology on smooth ∞-stacks with coefficients in ∞-Lie algebroid valued differential forms?

• What is the nonabelian Chern character – the Chern character of ∞-Lie groupoids
  with coefficients in deRham cohomology on ∞-stacks

We show that both these concepts are realized in terms of a structure that assigns to each ∞-stack $X$ its path ∞-groupoid $\Pi (X)$: the $\mathrm{\infty }$-stack whose k-morphisms are generated from the original $k$-morphisms in $X$ together with the $k$-dimensional smooth paths in $X$. This is a generalization of the construction of the fundamental groupoid of a topological space.

A morphism $\Pi (X)\to A$ out of the path $\mathrm{\infty }$-groupoid is encodes flat parallel transport along paths in $X$ with coefficients in $A$: an $A$-valued local system. This defines a cocycle in flat differential $A$-cohomology of $X$. Under suitable conditions such parallel transport is already entirely fixed by its restriction to the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)\subset \Pi (X)$ of $X$. The restricted morphism then fators through the ∞-Lie algebroid $𝔞$ of $A$ and the morphism ${\Pi }^{\inf }(X)\to 𝔞$ represents flat ∞-Lie algebroid valued differential forms constituting a cocycle in nonabelian deRham cohomology.

The canonical inclusion $X\hookrightarrow \Pi (X)$ of any ∞-stack into its path $\mathrm{\infty }$-groupoid gives rise to relative cohomology on $\Pi (X)$ relative to $X$. In terms of this the nonabelian Chern character map is defined as a map from $A$ cohomology to flat differential cohomology with real coefficients, hence to deRham cohomology.

Details

The discussion of technical details begins at

• theory of differential nonabelian cohomology.

History and references

The history of the eventual unravelling the general abstract nonsense conceptual underpinning of differential cohomology may roughly be divided into approaches that focused on flat differential cohomology, and those that considered general differential cocycles.

Flat differential cohomology

The flat case is naturally the one easier to grasp, and has accordingly earlier been considerably abstracted and generalized. The notion of Grothendieck connection and deRham descent and more generally the notion of local systems is a direct precursor of the notion of cocycles on the (infinitesimal) path ∞-groupoid.

Abelian differential cohomology

For the non-flat case a considerable insight has been the definition of differential refinements of generalized (Eilenberg-Steenrod) cohomology theories by Hopkins and Singer. This introduced the crucual notion that differential cocycles are objects in the homotopy fiber of a Chern character map. The generalization of this to nonabelian cohomology, however, has long been unclear.

Non-flat parallel transport

Instead, in a third historical thread there was an attempt to fully capture the notion of non-flat parallel transport directly. This development started out looking promising, but then ran into an unexpected problem: that of fake flatness . Pondering the resolution of this conundrum has finally led us to the observation that also nonabelian differential cohomology has to be conceived as a Chern-character twisted cohomology.

Here an outline of the history of the development of the notion of non-flat higher parallel transport.

It is an old observation that on connected spaces, principal bundles with connection are entirely encoded by the parallel transport which they induce on closed loop

• U. S., History of understanding bundles with connection using parallel transport around loops.

Motivated by this it was later observed that similarly an abelian gerbe with connection on a simply connected space is entirely encoded in the parallel surface transport that it induces on spheres:

• Roger Picken and Marco Mackaay, Holonomy and parallel transport for abelian gerbes (arXiv)

John Baez noticed that these facts suggest that the proper formulation of bundles and higher bundles with connection should be in terms of smooth parallel transport $n$-functors ${\Pi }_n(X)\to A$ that send n-categories of n-dimensional paths in a space $X$ to some smooth coefficient n-category $A$.

• John Baez, Categorified gauge theory (web)

At that point the motivation for this very natural definition was mainly formal, while the relevance of higher nonabelian parallel transport for physics was felt to be compelling but remained somewhat unclarified:

• John Baez, Higher Yang-Mills theory (arXiv)

• Girelli, Pfeiffer, Higher gauge theory – differential versus integral formulation (arXiv)

It took a bit longer for the full formalism to be developed and find its natural form.

While a description of nonabelian differential cocycles in degree 2, first given in

• J. Baez, U. S. Higher gauge theory (arXiv)

was available early on, accounts of all technical and some conceptual details appeared later.

In

• U.S. K. Waldorf, Parallel transport and functors (arXiv)

the full functorial parallel transport formulation in degree 1, i.e. for ordinary bundles with connection was described.

Based on this the description in degree 2 appeared in

• U. S., K. Waldorf, Smooth functors vs. differential forms (arXiv)

• U. S., K. Waldorf, Connections on non-abelian gerbes and their holonomy (arXiv)

This used the theory of concrete generalized smooth spaces developed in

• J. Baez, A. Hoffnung, Convenient categories of smooth spaces (arXiv)

The same description of degree 2 differential cocycle in a slightly different technical setup appeared in

• J. F. Martins, R. Picken, On 2-dimensional holonomy (arXiv)

In all these approaches so far what was considered was realled the cohomology of a co-skeleton truncation ${\Pi }_n(X)$ of the full fundamental ∞-groupoid $\Pi (X)={\Pi }_{\mathrm{\infty }}(X)$. As a result, the corresponding differential cocycles were non-flat only in degree $n$, but flat in lower degrees.

The generalization of differential nonabelian cohomology to the properly non-flat case – the “curvature-twisted flat case” – was described in terms not of Lie ∞-groupoids but in terms of the $L_{\mathrm{\infty }}$-algebras corresponding to them under higher Lie theory in

• H. Sati, U. S., J. Stashaff, $L_{\mathrm{\infty }}$-algebra connections and applications to String- and Chern-Simons $n$-transport (arXiv)

This article also identified the relevant classes of examples of higher nonabelian differential cocycles in physics: based on the insight in

• A. Henriques, Integrating $L_{\mathrm{\infty }}$-algebras (arXiv)

• J. Baez, A. Crans, U. S., D. Stevenson, From loop groups to 2-groups (arXiv)

that the String Lie 2-algebra considered in

• J. Baez, A. Crans, Lie 2-algebras (arXiv)

controls string structures and hence the Green-Schwarz anomaly cancellation mechanism in heterotic string theory, it was realized that this and related phenomena are examples of twisted nonabelian differential cocycles: a lift of a differential $G$-cocycle for $G$ some higher group to the next higher connected universal cover of $G$ is obstructed by a higher differential Chern-Simons cocycle, and the corresponding twisted lifts of differential cocycles are the higher gauge background field data satisfying the required anomaly cancellation conditions.

After an identification of the physics of this phenomenon in the next higher degree (which is 7) in

• H. Sati, U. S., J. Stasheff, Fivebrane structures (arXiv)

this story is developed in more detail in

• H. Sati, U. S., J. Stasheff, Twisted differential String- and Fivebrane structures, (arXiv)