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differential cohomology

Idea

Differential cohomology is a refinement of ordinary cohomology such that a differential cocycle is to its underlying ordinary cocycle as a bundle with connection is to its underlying bundle.

The details of a formulation of differential cohomology depend on how the generalized cohomology theory itself is formulated.

The best known version of differential cohomology is a differential refinement of generalized cohomology in the sense of the generalized Eilenberg–Steenrod axioms. This is a stable version of generalized cohomology.

A differential refinement of non-stable, i.e. nonabelian cohomology is developed here.

Differential stable cohomology

A standard definition of differential cohomology is in terms of a homtopy fiber product of a generalized Eilenberg–Steenrod cohomology theory with the complex of differential forms over real cohomology:

Let ${\Gamma }^{•}$ be a generalized cohomology theory in the sense of the generalized Eilenberg–Steenrod axioms, and let ${\Gamma }^{•}\to H^{•}(-,ℝ)\otimes {\Gamma }^{•}(*)$ be a morphism to real singular cohomology with coefficients in the ring of $\Gamma$-cohomology of the point. Then the differential refinement of ${\Gamma }^q$, the degree $q$differential $\Gamma$-cohomology is the homotopy pullback ${\overline{\Gamma }}^{•}$ in

$$\begin{array}{ccc}{\overline{\Gamma }}^{•}(-)& \stackrel{F}{\to }& {\Omega }^{\ge q}(-)\otimes {\Gamma }^{•}(*)\\ {\downarrow }^{\mathrm{cl}}& & \downarrow \\ {\Gamma }^{•}(-)& \stackrel{\mathrm{ch}}{\to }& Z^{•}(-,ℝ)\otimes {\Gamma }^{•}(*)\end{array},$$\array{ \bar \Gamma^\bullet(-) &\stackrel{F}{\to}& \Omega^{\geq q}(-)\otimes \Gamma^\bullet(*) \\ \downarrow^{cl} && \downarrow \\ \Gamma^\bullet(-) &\stackrel{ch}{\to}& Z^\bullet(-, \mathbb{R}) \otimes \Gamma^\bullet(*) } \,,

where

• $Z^{•}(-,ℝ)$ is the complex underlying real singular cohomology $H^{•}(-,ℝ)$

• ${\Omega }^{•}(-,ℝ)$ is the complex underlying deRham cohomology

• ${\Omega }^{•}(-)\to H^{•}(-,ℝ)$ is deRham theorem morphism

• $\mathrm{ch}:{\Gamma }^{•}(-)\to Z^{•}(-,ℝ)\otimes {\Gamma }^{•}(*)$ is the Chern character map.

There are variations of this definition, with some technical differences in the assumptions. See the description below.

Examples

• Differential integral cohomology ${\overline{H}}^{•}(-,\mathbb{Z})$ is modeled by

  • Deligne cohomology;

  • Cheeger-Simons differential character?s;

  • higher circle bundle gerbes with connection;

• apart from that people studied mainly differential K-theory.

In physics differential cocycles model gauge fields.

• Cocycles in ordinary differential cohomology (e.g. Deligne cohomology) model

  • in degree 2: the electromagnetic field

  • in degree 3: the Kalb-Ramond field

  • in degree 4: the supergravity C-field

• Cocycles in differential K-theory model the

  • RR-field .

For $c\in {\overline{\Gamma }}^{•}(X)$ a differentia cocycle representing a gauge, one says that

• its image $F(c)$ in differential forms is the corresponding field strength;

• its image $\mathrm{cl}(c)$ in non-differential cohomology is the “topological twist” of the gauge field. In special cases this can be identified with magnetic charge.

Differential cohomology following Bunke–Schick

The following theory of differential cohomology (also called smooth cohomology) is developed and used in the work of Ulrich Bunke and Thomas Schick?. Contrary to the above, it does not take the notion of homotopy limit as fundamental, but instead characterizes the universality of the above commuting diagram by other means. On the other hand, this means that their axiomatization at the moment only capture cohomology classes, not the representing cocycles. It is sort of known that for various applications specific cocycle representatives do play an important role, and one may imagining refining to discussion below eventually to accomodate for that.

• idea:

  • combine cohomology + differential forms

main diagram

$$\begin{array}{ccc}\hat{H}(M)& \stackrel{I}{\to }& H^{•}(M)\\ {\downarrow }^R& & \downarrow \\ {\Omega }_{d=0}^{•}(M)& \stackrel{}{\to }& H_{\mathrm{dR}}^{•}(M)\simeq H^{•}(M,ℝ)\end{array}$$\array{ \hat H(M) &\stackrel{I}{\to}& H^\bullet(M) \\ \downarrow^{R} && \downarrow \\ \Omega^\bullet_{d=0}(M) &\stackrel{}{\to}& H^\bullet_{dR}(M) \simeq H^\bullet(M,\mathbb{R}) }

so differential cohomology ${\hat{H}}^{•}(M)$ combines the ordinary cohomology $H^{•}(M)$ with a differential form representative of its image in real cohomology.

• $I$ projects a differential cohomology to its underlying ordinary cohomology class;

• $R$ send the differential cohomology class to its curvature differential form data

we want an exact sequence

$$\begin{array}{ccccccc}{H}^{•-1}(M)& \stackrel{\mathrm{ch}}{\to }& {\Omega }^{•-1}(M)/\mathrm{im}(d)& \stackrel{d}{\to }& \hat{H}(M)& \stackrel{I}{\to }& H^{•}(M)\to 0\\ & & & {}_d\searrow & {\downarrow }^R\\ & & & & {\Omega }_{d=0}^{•}(M)\end{array}$$\array{ H^{\bullet-1}(M) &\stackrel{ch}{\to}& \Omega^{\bullet-1}(M)/{im(d)} &\stackrel{d}{\to}& \hat H(M) &\stackrel{I}{\to}& H^\bullet(M) \to 0 \\ &&& {}_{d}\searrow & \downarrow^R \\ &&&& \Omega^\bullet_{d=0}(M) }

Definition

Given cohomology theory $E^{•}$, a smooth refinement ${\hat{E}}^{•}$ is a functor $\hat{E}:\mathrm{Diff}\to \mathrm{Grps}$ with transformations $I,R$ such that

$$\begin{array}{ccc}\hat{E}(M)& \stackrel{I}{\to }& E^{•}(M)\\ {\downarrow }^R& & \downarrow \\ {\Omega }_{d=0}^{•}(M,V)& \stackrel{}{\to }& E_{\mathrm{dR}}^{•}(M)\simeq E^{•}(M,ℝ)\end{array}$$\array{ \hat E(M) &\stackrel{I}{\to}& E^\bullet(M) \\ \downarrow^{R} && \downarrow \\ \Omega^\bullet_{d=0}(M, V) &\stackrel{}{\to}& E^\bullet_{dR}(M) \simeq E^\bullet(M,\mathbb{R}) }

where

$V=E^{•}(\mathrm{pt})\otimes ℝ$

is the graded non-torsion cohomology of $E$ on the point (so now all the gradings above denote total grading)

and such that there is a transformation

$$a:{\Omega }^{•-1}(M)/\mathrm{im}(d)\to {\hat{E}}^{*}(M)$$a : \Omega^{\bullet -1}(M)/{im(d)} \to \hat E^*(M)

that gives the above kind of exact sequence.

Definition

If $E^{*}$ is multiplicative, we say ${\hat{E}}^{*}$ is multiplicative with product $\vee$ if $\hat{E}$ takes values in graded rings and the transformations are compatible with multiplicative structure, where

$$a(\omega )\vee x=a(\omega \wedge R(x))$$a(\omega) \vee x = a(\omega \wedge R(x))

Definition

$\hat{E}$ has $S^1$-integration if there is a natural (in $M$) transformation

$$\int :{\hat{E}}^{*}(M\times S^1)\to {\hat{E}}^{•-1}(M)$$\int : \hat E^*(M \times S^1) \to \hat E^{\bullet -1}(M)

compatible with $\int$ of forms and for $E$ it is given by the suspension isomorphism

$$\int \circ p^{*}=0$$\int \circ p^* = 0

for $p:M\times S^1\to M$ and

$$\int \circ (\mathrm{id}\times (z\mapsto \overline{z}){)}^{*}=-\int$$\int \circ ( id \times (z \mapsto \bar z) )^* = - \int

Remark

Ordinary cohomology theories are supposed to be homotopy invariant, but differential forms are not, so in general the differential cohomology is not.

Lemma

Given $\hat{E}$ a smooth cohomology theory. The homotopy formula:

given $h:M\times [\mathrm{0,1}]\stackrel{\mathrm{smooth}}{\to }N$ a smooth homotopy we have

$$h_1^{*}(X)-h_0^{*}(X)=a({\int }_{M\times [\mathrm{0,1}]/M}{h}^{*}(R(x)))$$h^*_1(X) - h^*_0(X) = a( \int_{M \times [0,1]/M} h^*(R(x)))

Corollary

$\ker (R)$ (i.e. flat cohomology) is a homotopy invariant functor.

Definition

$\hat{H}\mathrm{flat}:=\ker (R)$

Proof of lemma

It suffices to show

$${\iota }_1^{*}(x)-{\iota }_0^{*}(x)=a({\int }_{M\times [\mathrm{0,1}]/M}R(x))$$\iota_1^*(x) - \iota_0^*(x) = a(\int_{M\times [0,1]/M} R(x))

for all $x\in \hat{E}(M\times [\mathrm{0,1}])$.

Observe if $x=p^{*}y$ the left hand side vanishes, $\int R(p^{*}y)=0$.

For general $x$ $\exists y\in jhatE(M)$; $x-p^{*}(y)=a(\omega )$ $\omega \in \Omega (M\times [\mathrm{0,1}])$.

Stokes’ theorem gives $i_1^{*}\omega -i_0^{*}\omega ={\int }_{[\mathrm{0,1}]}d\omega$ $=\int R(a(\omega ))=\int R(x-p^{*}\omega )=\int R(x)$.

On the other hand

$$i_1^{*}(x)-i_0^{*}(x)=i_1^{*}(a(\omega ))-i_0^{*}(a(\omega ))=a(\int R(x))$$i^*_1(x) - i^*_0(x) = i^*_1(a(\omega)) - i^*_0(a(\omega)) = a(\int R(x))

A calculation: ${\hat{H}}_{\mathrm{flat}}^1(\mathrm{pt})={\hat{H}}^1(\mathrm{pt})=ℝ/\mathbb{Z}={\hat{K}}^1(\mathrm{pt})$.

Theorem (Hopkins–Singer)

For each generalized cohomology theory $E^{*}$ a differential version ${\hat{E}}^{*}$ as in the above definition does exist.

Moreover ${\hat{E}}_{\mathrm{flat}}^{*}=Eℝ/{\mathbb{Z}}^{•-1}$.

Remark

It’s not evident how to obtain more structure like multiplication.

Theorem

Using geometric models, multiplicative smooth extensions with $S^1$-integration are constructed for

• K-theory (Bunke–Schick)

• MU-bordisms (unitary bordisms)
  (Bunke–Schröder–Schick–Wiethaupts; and from there Landweber exact cohomology theories)

Uniqueness theorem (Bunke–Schick)

(Simons–Sullivan proved this for ordinary integral cohomology.)

Assume $E^{*}$ is rationally even, meaning that

$$E^k(\mathrm{pt})\otimes \mathbb{Q}=0\mathrm{for}\mathrm{odd}k$$E^k(pt)\otimes \mathbb{Q} = 0 \;\; for odd k

plus one further technical assumption.

Then any two smooth extensions ${\hat{E}}^{*}$, ${\tilde{E}}^{*}$ are naturally isomorphic.

If required to be compatible with integration the ismorphism is unique.

If $\hat{E},\tilde{E}$ are multiplicative, then this isomorphism is, as well.

Example

If we don’t require compatibility with $S^1$-integration, then there are “exotic” abelian group structures on ${\hat{K}}^1$.

References

A detailed discussion of the axiomatization of differential stable cohomology is

• Mike Hopkins and I. Singer, Quadratic Functions in Geometry, Topology,and M-Theory (arXiv)

Based on this Dan Freed interpreted large classes of gauge fields in physics in terms of differential stable cohomology in

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

The differential refinement of K-theory was and is studied in a series of articles by Bunke and Schick. See for instance

• Ulrich Bunke, Differential cohomology in geometry and analysis (pdf slides)

and many more…

talk notes

• the talks by Dan Freed, Schick, Ulrich Bunke and Schreiber at Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology. Much of the above material is taken from Thomas Schick’s lecture there.

blog discussion

• (Generalized) Differential Cohomology and Lie Infinity-Connections

Differential non-abelian cohomology

for the moment, see Differential Nonabelian Cohomology