theory of differential nonabelian cohomology



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.


The most general notion of cohomology H(X,A)H(X,A) of an object XX with coefficients in an object AA supposes that XX and AA are both objects of an (∞,1)-topos H\mathbf{H} and is defined by

H(X,A):=Ho H(X,A):=π 0H(X,A), H(X,A) := Ho_{\mathbf{H}}(X,A) := \pi_0 \mathbf{H}(X,A) \,,

where Ho HHo_{\mathbf{H}} is the homotopy category of H\mathbf{H}, whose hom-sets are the connected components of the ∞-groupoid H(X,A)\mathbf{H}(X,A) of maps, homotopies between maps, homotopies between homotopies etc., from XX to AA.

Since AA could be but is not required to be a (connective) spectrum this is more general than what is called generalized (Eilenberg-Steenrod) cohomology: both in that

and in that

A standard example of an object in parameterized nonabelian cohomology is a nonabelian gerbe. More generally, nonabelian cohomology classifies principal ∞-bundles.

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.

differentialnonabeliancohomology refines nonabeliancohomology generalizes generalizes differentialcohomology refines generalized(EilenbergSteenrod)cohomology \array{ differential nonabelian cohomology &\stackrel{refines}{\to}& nonabelian cohomology \\ \downarrow^{generalizes} && \downarrow^{generalizes} \\ differential cohomology &\stackrel{refines}{\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.

The path ∞-groupoid

In order to extract differential cohomology in the context given by some (∞,1)-topos H\mathbf{H} we need to have a notion of parallel transport along paths in the objects of H\mathbf{H} . This is encoded by assigning to each object XX its path ∞-groupoid Π(X)\Pi(X). Morphisms Π(X)A\Pi(X) \to A enocode flat AA-valued parallel transport on XX or equivalently AA-valued local systems on XX.

Structurally, regarding the path ∞-groupoid assignment

Π():HH \Pi(-) : \mathbf{H} \to \mathbf{H}

makes the (∞,1)-topos H\mathbf{H} into a structured (∞,1)-topos. This assignment has a right adjoint

HH:() flat. \mathbf{H} \leftarrow \mathbf{H} : {(-)}_{flat} \,.

Flat differential cohomology

In the presence of a notion of path ∞-groupoid we take flat differential AA-valued cohomology to be the cohomology with coefficients in an object A flatA_{flat} in the image of this right adjoint, and write

H flat(X,A):=H(Π(X),A)H(X,A flat). H_{flat}(X,A) := H(\Pi(X),A) \simeq H(X,A_{flat}) \,.

A cocycle in flat differential AA-cohomology may be thought of as a local system with coefficients in AA.

There is a natural morphism

XΠ(X) X \hookrightarrow \Pi(X)

that includes each object as the collection of 0-dimensional paths into its path ∞-groupoid. This induces correspondingly a natural morphism of coefficient objects A flatA A_{flat} \to A.

Lifting an AA-cocycle XAX \to A through this morphism to a flat differential AA-cocycle means equipping it with a flat connection.

Differental cohomology with curvature classes

We identify general non-flat differential nonabelian cocycles with the obstruction to the existence of the lift through A flatAA_{flat} \to A from bare AA-cohomology to flat differential AA-cohomology.

There are two flavors of this obstruction theory whose applicability depeends on whether AA is group-like (once deloopable) or more generally nonabelian.

In the general case of non-group-like AA we shall use a [[nLab:Chern-character] morphism to approximate AA 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.

Therefore we now first indicate the theory for the case of grouplike coefficients and then describe the theory with general coefficients in terms of that.

With grouplike coefficients

We show that when AA grouplike in that its delooping BA\mathbf{B}A exists, the morphism A flatAA_{flat} \to A fits into a fibration sequence

A flatABA dR A_{flat} \to A \to \mathbf{B}A_{dR}

where BA dR\mathbf{B}A_{dR} is the coefficient for nonabelian deRham cohomology with coefficients in BA\mathbf{B}A: a cocycle XBA dRX \to \mathbf{B}A_{dR} is a flat differential BA\mathbf{B}A-cocycle whose underlying ordinary AA-cocycle is trivial.

This being a fibration sequence means that the obstruction to lifting an AA-cocycle XAX \to A to a flat differential AA-cocycle is the BA dR\mathbf{B}A_{dR}-cocycle given by the composite map XABA dRX \to A \to \mathbf{B}A_{dR}. The class of this BA\mathbf{B}A-cocycle we call the curvature characteristic class of the original AA-cocycle.

If this obstruction does not vanish but is given by some fixed curvature characteristic class PP, there is a general notion of twisted cohomology that encodes the PP-twisted-flat (namely: non-flat) differential cocycles.

This is finally our definition of differential nonabelian cohomology with grouplike coeffiecients.

For AA a grouplike object in H\mathbf{H} and for [P]H(X,BA dR)[P] \in H(X,\mathbf{B}A_{dR}) a fixed curvature characteristic class, the differential AA-cohomology with curvature characteristic PP is the homotopy pullback

H¯ [P](X,A) * *P H(X,A) H(X,BA dR). \array{ \bar \mathbf{H}_{[P]}(X,A) &\to& {*} \\ \downarrow && \;\downarrow^{{*} \mapsto P} \\ \mathbf{H}(X,A) &\to& \mathbf{H}(X,\mathbf{B}A_{dR}) } \,.

We show that the objects in H¯ [P](X,A)\bar \mathbf{H}_{[P]}(X,A) correspond to diagrams in SPSh(C)SPSh(C) of the form

Y A underlyingcocycle firstEhresmanncondition Π(Y) EA connection secondEhresmanncondition Π(X) F BA characteristicforms \array{ Y &\to& A &&& underlying cocycle \\ \downarrow && \downarrow &&& first Ehresmann condition \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A &&& connection \\ \downarrow && \downarrow &&& second Ehresmann condition \\ \Pi(X) &\stackrel{F}{\to}& \mathbf{B}A &&& characteristic forms }

where YXY \to X is the Cech nerve of a given cover.

Here the interpretation of each layers is as indicated, as discussed in more detail after the next subsection.

With general coefficients

If AA is not grouplike, one may use grouplike approximations to AA and proceed with these through the above constructions. One drastic but useful grouplike approximation to any AA is given by an integral Chern character morphism

ch:A iB n iR//Z. ch : A \to \prod_i \mathbf{B}^{n_i} R//Z \,.

We sow that from this one obtains along the above lines an object BA ch\mathbf{B}A_{ch} equipped with a morphism ABA chA \to \mathbf{B}A_{ch} such that the induced morphism on cohomology

ch:H(,A)H(,BA ch) ch : \mathbf{H}(-,A) \to \mathbf{H}(-, \mathbf{B}A_{ch})

generalizes the Chern character morphism in abelian cohomology.

Proceeding with this morphism as a substitute for the non-existing ABA dRA \to \mathbf{B}A_{dR} 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 BA\mathbf{B}A for either the delooping of AA, if it exists, or ellse for the delooping of the codomain of the Chern character map of AA.

Ehresmann \infinity-connection

We show how the above differential cocycles constitute an \infty-groupoidal version of the notion of Ehresmann connection.

This is achieved by noticing that every cocycle XAX \to A trivializes on the total space PXP \to X of the principal ∞-bundle PP that it classifies. On PP, we have the vertical path ∞-groupoid? Π vert(P)\Pi_{vert}(P) of fiberwise paths. We show that every differential cocycle encoded by a diagram as above gives rise to a diagram

Π vert(P) A flatverticalAvalueddifferentialform firstEhresmanncondition Π(P) EA Avaluedconnectionformontotalspace secondEhresmanncondition Π(X) P BA characteristicformsonbasespace \array{ \Pi_{vert}(P) &\to& A &&& flat vertical A-valued differential form \\ \downarrow && \downarrow &&& first Ehresmann condition \\ \Pi(P) &\stackrel{\nabla}{\to}& \mathbf{E}A &&& A-valued connection form on total space \\ \downarrow && \downarrow &&& second Ehresmann condition \\ \Pi(X) &\stackrel{P}{\to}& \mathbf{B}A &&& characteristic forms on base space }

where each horizontal morphism is a cocycle in nonabelian deRham cohomology.

A principal ∞-bundle PP equipped with such nonabelian deRham cocycle data we call an Ehresmann ∞-connection as it generalizes the notion of Ehresmann connection on ordinary principal bundles.

Its expression in terms of ∞-Lie algebroid valued differential forms on the total space of the principal ∞-bundle is given by the next step.

Cartan-Ehresmann \infinity-connection

The above is still a generalization of the notion of Ehresmann connection to ambient (∞,1)-toposes that need not necessarily have an ordinary notion of differential forms.

To obtain that we assume for the following that H\mathbf{H} is actually a smooth (∞,1)-topos. In that context we have a notion of ∞-Lie theory.

This allows to replace in the above diagram all ∞-Lie groupoids appearing with their infinitesimal approximations: their ∞-Lie algebroids.

Under these translations the above diagram characterizing an Ehresmann ∞-connection translates into a diagram of graded commutative dg-algebras

Ω vert (P) A vert CE(𝔞) flatvertical𝔞valueddifferentialform firstEhresmanncondition Ω (P) (A,F A) W(𝔞) 𝔞valuedconnectionformontotalspace secondEhresmanncondition Ω (X) P(F A) inv(𝔞) curvaturecharacteristicformsonbasespace \array{ \Omega^\bullet_{vert}(P) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{a}) &&& flat vertical \mathfrak{a}-valued differential form \\ \uparrow && \uparrow &&& first Ehresmann condition \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& W(\mathfrak{a}) &&& \mathfrak{a}-valued connection form on total space \\ \uparrow && \uparrow &&& second Ehresmann condition \\ \Omega^\bullet(X) &\stackrel{P(F_A)}{\leftarrow}& inv(\mathfrak{a}) &&& curvature characteristic forms on base space }

This we call a Cartan-Ehresmann ∞-connection. This finally constitutes concrete ∞-Lie algebroid valued differential forms data on a principal ∞-bundle. These L L_\infty-algebra connections have been discussed in

  • H. Sati, U. S., J. Stasheff, L L_\infty-algebra connections and their applications to String- and Chern-Simons nn-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)

Details of the theory

This concludes the extended abstract . For details of the theory proceed with

Revised on January 8, 2010 00:22:47 by Toby Bartels (