Schreiber theory of differential nonabelian cohomology

Abstract

We describe a theory that

• generalizes the notion of differential cohomology from abelian cohomology to nonabelian cohomology;

• generalizes the notion of connection on a bundle together with the notion of parallel transport from principal bundles to higher gerbes and principal ∞-bundles.

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.

Idea

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

$$H(X,A) := Ho_{\mathbf{H}}(X,A) := \pi_0 \mathbf{H}(X,A) \,,$$

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

Since $A$ 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

• $X$ and $A$ may have more structure than just topological spaces, notably they may have smooth structure in that they are parameterized over smooth test spaces.

  Such parameterized objects are called ∞-stacks– for instance smooth ∞-stacks.

and in that

• $A$ need not be abelian or stable . It may in particular be an arbitrary homotopy n-type. More generally, $A$ 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.

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.

$$\array{ differential nonabelian cohomology &\stackrel{refines}{\to}& nonabelian cohomology \\ \downarrow^{generalizes} && \downarrow^{generalizes} \\ differential cohomology &\stackrel{refines}{\to}& generalized (Eilenberg-Steenrod) cohomology }$$

Plan

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 $\mathbf{H}$ we need to have a notion of parallel transport along paths in the objects of $\mathbf{H}$ . This is encoded by assigning to each object $X$ its path ∞-groupoid $\Pi(X)$. Morphisms $\Pi(X) \to A$ enocode flat $A$-valued parallel transport on $X$ or equivalently $A$-valued local systems on $X$.

Structurally, regarding the path ∞-groupoid assignment

$$\Pi(-) : \mathbf{H} \to \mathbf{H}$$

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

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

Flat differential cohomology

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

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

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

There is a natural morphism

$$X \hookrightarrow \Pi(X)$$

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

Lifting an $A$-cocycle $X \to A$ through this morphism to a flat differential $A$-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 $A_{flat} \to A$ from bare $A$-cohomology to flat differential $A$-cohomology.

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

In the general case of non-group-like $A$ we shall use a [[nLab:Chern-character] morphism to approximate $A$ 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 $A$ grouplike in that its delooping $\mathbf{B}A$ exists, the morphism $A_{flat} \to A$ fits into a fibration sequence

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

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

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

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

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

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

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

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

$$\array{ Y &\to& A &&& underlying cocycle \\ \downarrow && \downarrow &&& first Ehresmann condition \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A &&& connection \\ \downarrow && \downarrow &&& second Ehresmann condition \\ \Pi(X) &\stackrel{F}{\to}& \mathbf{B}A &&& characteristic forms }$$

where $Y \to X$ 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 $A$ is not grouplike, one may use grouplike approximations to $A$ and proceed with these through the above constructions. One drastic but useful grouplike approximation to any $A$ is given by an integral Chern character morphism

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

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

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

generalizes the Chern character morphism in abelian cohomology.

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

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

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

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

A principal ∞-bundle $P$ 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 $\mathbf{H}$ 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

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

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

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

• differential cohomology?

  • differential cohomology - with grouplike coefficients?

  • differential cohomology - with general coefficients

• Examples

  • principal bundles with connection and differential K-theory

  • principal 2-bundles with connection

• Applications

  • Background fields in twisted differential nonabelian cohomology

  • differential twisted String and Fivebrane structures

  • D'Auria-Fre formulation of supergravity