Differential K-theory. A survey.

Differential K-theory. A survey.

Ulrich Bunke
Johannes-Kepler-Universität Regensburg
Deutschland
ulrich.bunke@mathematik.uni-regensburg.de
Thomas Schick
Georg-August-Universität Göttingen
Deutschland
e-mail: schick@uni-math.gwdg.de
www: http://www.uni-math.gwdg.de/schick
Fax: ++49 -551/39 2985
Abstract

Generalized differential cohomology theories, in particular differential K-theory (often called “smooth K-theory”), are becoming an important tool in differential geometry and in mathematical physics.

In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer family index theorem for differential K-theory.

1 Introduction

The most classical differential cohomology theory is ordinary differential cohomology with integer coefficients. It has various realizations, e.g. as smooth Deligne cohomology (compare [18]) or as Cheeger-Simons differential characters [32]. In the last decade, differential extensions of generalized cohomology theories, in particular of K-theory, have been studied intensively. In part, this is motivated by its application in mathematical physics, for the description of fields with quantization anomalies in abelian gauge theories, suggested by Freed in [36], compare also [44].

The basic idea is that a differential cohomology theory should combine cohomological information with differential form information. More precisely, given a generalized cohomology theory together with a natural transformation , to cohomology with coefficients in a graded real vector space , and using an appropriate setup one can define the differential refinement of as a homotopy pullback

 ^E(X)I−−−−→E(X)⏐⏐↓R⏐⏐↓chΩd=0(X;N)Rham−−−−→H(X;N).

The natural transformations (the underlying cohomology class) and (the characteristic closed differential form) are essential parts of the picture. With slight abuse of notation, we call the curvature homomorphism. This is a bit of a misnomer, as in a geometric situation will be determined by the honest curvature, but not vice versa. is not a generalized cohomology theory and not meant to be one: it contains differential form information and as a consequence is not homotopy invariant.

If is ordinary integral cohomology, is just induced by the inclusion of coefficients . For K-theory, the situation we are mainly discussing in this article, is the ordinary Chern character.

The flat part of is defined as the kernel of the curvature morphism:

 ^Eflat(X):=ker(R:^E(X)→Ω(X;N)) .

It turns out that is a cohomology theory, usually just , the generalized cohomology with -coefficients with a degree shift: . An original interest in differential K-theory (before it even was introduced as such) was its role as a geometric model for . Karoubi in [46, Section 7.5] defined using essentially the flat part of a cycle model for , compare also Lott [54, Definition 5, Definition 7] where also is introduced. Homotopy theory provides a universal construction of for a generalized cohomology theory . However, this is in general hard to combine with geometry.

K-theory is the home for index theory. The differential K-theory (in particular its different cycle models) and also its flat part naturally are the home for index problems, taking more of the geometry into account. Indeed, in suitable models it is built into the definitions that geometric families of Dirac operators, parameterized by , give rise to classes in , where is the parity of the dimension of the fiber. A submersion with closed fibers with fiberwise geometric -structure (the precise meaning of geometry will be discussed below) is oriented for differential K-theory and one has an associated push-forward

 ^p!:^K∗(X)→^K∗−d(Y)

(with ). The same data also gives rise to a push-forward in Deligne cohomology

 ^p!:^H∗(X)→^H∗−d(Y) .

There is a unique lift of the Chern character to a natural transformation

Here, the right hand side is the differential extension of and one of the main results of [26] is a refinement of the classical Riemann-Roch theorem to a differential Riemann-Roch theorem which identifies the correction for the compatibility of with the push-forwards. In [39], Bismut superconnection techniques are used to define the analytic index of a geometric family: it can be understood as a particular representative of the differential K-theory class of a geometric family as above, determined by the analytic solution of the index problem. Moreover, they develop a geometric refinement of the topological index construction of Atiyah-Singer [3] based on geometrically controlled embeddings into Euclidean space (which does not require deep spectral analysis) and prove that topological and analytic index in differential K-theory coincide.

Finally, we observe that, in suitable special situations, we can easily construct classes in differential K-theory which turn out not to depend on the special geometry, but only on the underlying differential-topological data. Typically, these live in the flat part of differential K-theory and are certain (generalizations of) secondary index invariants. Examples are rho-invariants of the Dirac operator twisted with two flat vector bundles (and family versions hereof), or the -index of Lott [54] for a manifold whose boundary is identified with the disjoint union of copies of a given manifold .

Similar to smooth Deligne cohomology, there is a counterpart of differential K-theory in the holomorphic setting [41] and there is an arithmetic Riemann-Roch for these groups. This, however, will not be discussed in this survey.

1.1 Differential cohomology and physics

A motivation for the introduction of differential K-theory comes from quantum physics. The fields of abelian gauge theories are described by objects which carry the local field strength information of a closed differential form (assuming that there are no sources). Dirac quantization, however, requires that their de Rham classes lie in an integral lattice in de Rham cohomology. For Maxwell theory the field strength simply is a -form which is the curvature of a complex line bundle and therefore lies in the image of ordinary integral cohomology. For Ramond-Ramond fields in type II string theories it is a differential form of higher degree which lies in the image under the Chern character of K-theory, as suggested by [57, 37]. Indeed, Witten suggests that D-brane charges in the low energy limit of type IIA/B superstring theory are classified by K-theory. In this case, even if the field strength differential form is zero, the fields or D-brane charges can contain some global information, corresponding to torsion in K-theory.

It is suggested by Freed [36] that these Ramond-Ramond fields are described by classes in differential K-theory (or other generalized differential cohomology theories, depending on the particular physical model). Given a space-time background and a field represented by a class , this field contains the differential form information (as expected for an abelian gauge field). The field equations (generalizing Maxwell’s equations) require that if there are no sources (which we assume here). However, there is a quantization condition: the de Rham class represented by is not arbitrary, but lies in an integral lattice, namely in . Indeed, also contains the integral (and possibly torsion) information of the class . Finally, even and together don’t determine entirely, there is extra information, corresponding to a physically significant potential or holonomy. More precisely, is the configuration space with a gauge group action. Details of such a gauge field theory are studied e.g. in [50], where it is shown that the free part of is an obstruction to a global gauge fixing. Nonetheless, [50] proposes a partition function and among others computes the vacuum expectation value.

All discussed so far describes the situation without any background field or flux. However, such background fields are an important ingredient of the theory. Depending on the chosen model and the precise situation, a background field can be defined in many different ways. In the classical situation where fields are just given by differential forms, a background field is a closed -form . It creates an extra term in the field equations. Correspondingly, the relevant charges are even or odd forms (depending on the type of the theory) which are closed for the differential with . And they are classified up to equivalence by the -twisted de Rham cohomology

 H∗+ΩdR(X):=ker(dΩ)/im(dΩ).

When looking at charges in the presence of a background B-field (producing an H-flux) which are classified topologically by K-theory, we need to work with twisted K-theory, compare in particular [67, 17]. We will give a short introduction to twisted K-theory in Section 7.1. The role of twisted K-theory is discussed a lot in the case of T-duality. T-duality predicts an isomorphism of string theories on different background manifolds which are T-dual to each other, and in particular an isomorphism of the K-theory groups which classify the D-brane charges.

It turns out, however, that the topology of one of the partners in duality dictates a background B-field on the other, and the required isomorphism can only hold in twisted K-theory, compare e.g. [16, 23]. In those papers, mainly the topological classification of D-brane charges is considered. A new picture now arises when one wants to move to T-duality for Ramond-Ramond fields described by differential K-theory as explained above. One has to construct and study twisted differential K-theory. A first step toward this is carried out in [31]. Now, physicists try to understand T-duality at the level of Ramond-Ramond fields, compare e.g. [9] where the ideas are discussed explicitly without mathematical rigor. With mathematical rigor, the T-duality isomorphism in (twisted) differential K-theory has been worked out by Kahle and Valentino in [45]. We will describe these results in more detail in Section 7.4.

2 Axioms for differential cohomology

A fruitful approach to generalized cohomology theories is based on the Eilenberg-Steenrod axioms. It turns out that many of the basic properties of smooth Deligne cohomology and differential K-theory also are captured by a rather small set of axioms, proposed in [26, Section 1.2.2] (and motivated by [36]). Therefore, we want to base our treatment of differential K-theory on those axioms, as well.

The starting point is a generalized cohomology theory , together with a natural transformation , where is a graded coefficient -vector space. The two basic examples are

• , ordinary cohomology with integer coefficients, where and is induced by the inclusion of coefficients . More generally, can be replaced by any subring of , e.g. by .

• , K-theory, where is the usual Chern character, and with of degree . Multiplication with corresponds to Bott periodicity.

2.1 Definition.

A differential extension of the pair is a functor from the category of compact smooth manifolds (possibly with boundary) to -graded groups together with natural transformations

1. (curvature)

2. (underlying cohomology class)

3. (action of forms) .

Here 111This definition has to be modified in a generalization to non-compact manifolds! denote the smooth differential forms with values in , the usual de Rham differential and the space of closed differential forms ( stands for degree-shift by ).

The transformations are required to satisfy the following axioms:

1. The following diagram commutes

 ^E(X)I−−−−→E(X)⏐⏐↓R⏐⏐↓chΩd=0(X,N)Rham−−−−→H(X;N). (2.1)
2.  R∘a=d . (2.1)
3. is of degree .

4. The following sequence is exact:

 E∗−1(X)ch−→Ω∗−1(X,N)/im(d)a→^E∗(X)I→E∗(X)→0 . (2.1)

Alternatively, when dealing with K-theory one can and often (e.g. in [26]) does consider the whole theory as -graded with the obvious adjustments. Note that with we have natural and canonical isomorphism and . The associated -graded ordinary cohomology is therefore given by the direct sum of even or odd degree forms.

2.2 Corollary.

If is a differential extension of , then we have a second exact sequence

 E∗−1(X)ch−→H∗−1(X;N)a→^E∗(X)R→Ω∗d=0(X;N)×chE∗(X)→0, (2.2)

where is the pullback of abelian groups.

Proof.

This is a direct consequence of (2.1), (2.1) and (2.1). ∎

2.3 Definition.

Given a differential extension of a cohomology theory , we define the associated flat functor

 ^Eflat(X):=ker(R:^E(X)→Ωd=0(X;N)).
2.4 Remark.

The naturality of indeed implies that is a contravariant functor on the category of smooth manifolds. Actually, this functor by Corollary 2.7 is always homotopy invariant and extends to a cohomology theory in many examples, as we will discuss in Section 2.3. Typically, there is a natural isomorphism , but we still don’t know whether this is necessarily always the case (compare the discussion in [27, Section 5 and Section 7]).

The most interesting cases are not just group valued cohomology functors, but multiplicative cohomology theories, for example K-theory and ordinary cohomology. We therefore want typically a differential extension which carries a compatible product structure.

2.5 Definition.

Assume that is a multiplicative cohomology theory, that is a -graded algebra over , and that is compatible with the ring structures. A differential extension of is called multiplicative if together with the transformations is a differential extension of , and in addition

1. is a functor to -graded rings,

2. and are multiplicative,

3. for all and .

Deligne cohomology is multiplicative [18, Chapter 1], [35, Section 6], [20, Section 4]. In this paper, we will consider multiplicative extensions of -theory.

2.1 Variations of the axiomatic approach

Our list of axioms for differential cohomology theories seems particularly natural: it allows for efficient constructions and to derive the conclusions we are interested in. However, for the differential refinements of integral cohomology, a slightly different system of axioms has been proposed in [62]. The main point there is that the requirement of a given natural isomorphism between the flat part of Definition 2.3 and with coefficients in . It turns out that, for differential extensions of ordinary cohomology, both sets of axioms imply that there is a unique natural isomorphism to Deligne cohomology (compare [62] and [27, Section 7]). In particular, for ordinary cohomology they are equivalent. The corresponding result holds in general under extra assumptions, which are satisfied for K-theory, compare Section 2.3.

2.2 Homotopy formula

A simple, but important consequence of the axioms is the homotopy formula. If one differential cohomology class can be deformed to another, this formula allows to compute the difference of the two classes entirely in terms of differential form information. In a typical application, one will deform an unknown class to one which is better understood and that way get one’s hands on the complicated class one started with.

2.6 Theorem (Homotopy formula).

Let be a differential extension of . If and are the inclusions then

 i∗1x−i∗0x=a(∫[0,1]×B/BR(x)),

where denotes integration of differential forms over the fiber of the projection with the canonical orientation of the fiber .

Proof.

Note that, if for some then by naturality the left hand side of the equation is zero. Moreover, in this case so that by the properties of integration over the fiber the right hand side vanishes, as well.

In general, observe that is a homotopy equivalence, so that we always find with . Using surjectivity of , we find with , and then . Since the sequence (2.1) is exact, there is with . Stokes’ theorem applied to yields

 i∗1ω−i∗0ω=∫[0,1]×B/Bdω.

On the other hand, because of (2.1), . Substituting, we get

 ∫[0,1]×B/Bdω=∫[0,1]×B/BR(a(ω))=∫[0,1]×B/BR(x−p∗y)=∫[0,1]×B/BR(x)

and (using again vanishing of our expressions for )

 i∗1x−i∗0=i∗1a(ω)=i∗0a(ω)=a(i∗1ω−i∗0ω)=a(∫[0,1]×B/BR(x)).

2.7 Corollary.

Given a differential extension of , the associated flat functor of Definition 2.3 is homotopy invariant.

Proof.

Let be a homotopy between and . We have to show that . By functoriality, it suffices to show that , as and . This, however, follows immediately from Theorem 2.6 once . ∎

We have seen above that is a homotopy invariant functor. Ideally, it should extend to a generalized cohomology theory (compare the discussion of Section 2.1). For this, we need a bit of extra structure which corresponds to the suspension isomorphism, and which is typically easily available. We formulate this in terms of integration over the fiber for , originally defined in [27, Definition 1.3]. Note that the projection is canonically oriented for an arbitrary cohomology theory because the tangent bundle of is canonically trivialized. For a push-down in differential cohomology, in general we expect that one has to choose geometric data of the fibers, which again we can assume to be canonically given for the fiber . Orientations and push-forward homomorphisms for differential cohomology theories, in particular for differential K-theory are discussed in Section 5.

2.8 Definition.

We say that a differential extension of a cohomology theory has -integration if there is a natural transformation which is compatible with the transformations and the “integration over the fiber” as well as . In addition, we require that for each and for each , where is complex conjugation.

In [27, Corollary 4.3] we prove that in many situations, e.g. for ordinary cohomology or for K-theory, there is a canonical choice of integration transformation.

2.9 Theorem.

If is a multiplicative differential extension of and if is a torsion group, then has a canonical -integration as in Definition 2.8.

2.3 ^E∗flat as generalized cohomology theory ER/Z

Let be a generalized cohomology theory. In the present section we consider a universal differential extension , i.e. we take and let be the canonical transformation.

To there is an associated generalized cohomology theory ( with coefficients in ). It is constructed with the help of stable homotopy theory: the cohomology theory is given by a spectrum (in the sense of stable homotopy theory) , and is given by the spectrum , where is the Moore spectrum of the abelian group . is constructed in such a way that one has natural long exact sequences

 →E∗(X)→ER∗(X)→ER/Z∗(X)→E∗+1(X)→…

Note that for a finite -complex one can alternatively write with defined by smash product with .

In the fundamental paper [44], Hopkins and Singer construct a specific differential extension for any generalized cohomology theory . For this particular construction, one has by [44, (4.57)] a natural isomorphism

 ER/Z∗−1(X)→E∗flat(X).

However, this is a consequence of the particular model used in [44]. It is therefore an interesting question to which extend the axioms alone imply that is a generalized cohomology theory. Here, some extra structure about the suspension isomorphism seems to be necessary, implemented by the transformation “integration over ” of Definition 2.8. With a surprisingly complicated proof one gets [27, Theorem 7.11]:

2.10 Theorem.

If is a differential extension of with integration over , then has natural long exact Mayer-Vietoris sequences. It is equivalent to the restriction to compact manifolds of a generalized cohomology theory represented by a spectrum. Moreover, and are natural transformations of cohomology theories and one obtains a natural long exact sequence for each finite CW-complex

 →ER∗−1(X)a→^E∗flat(X)I→E∗(X)ch−→ER∗(X)→. (2.10)
Proof.

The long exact sequence (2.10) is not stated like that in [27] and we will need it below, therefore we explain how this is achieved. Because of (2.1), the restriction of to (realized as de Rham cohomology, i.e. as ) hits exactly . The exactness at therefore is a direct consequence of the exactness of (2.1). (2.1) also implies immediately that in (2.10). Because of (2.1), in (2.10). Finally, as is surjective, any can be written as with and such that , i.e.  for some . But then and , which implies that (2.10) is also exact at . ∎

2.11 Corollary.

Let be a differential extension of with integration over , or more generally assume that has natural long exact Mayer-Vietoris sequences. Assume that are closed submanifolds of codimension with boundary (and corners) such the interiors of and cover . Then one has a long exact Mayer-Vietoris sequence

 ^En+1flat(X0)i∘δ−→^En(X)→^En(X1)⊕^En(X2)→^En(X0)δ∘I−−→En−1(X)⋯

which continues to the right with the Mayer-Vietoris sequence for and to the left with the Mayer-Vietoris sequence for .

Proof.

The proof is an standard diagram chase, using the Mayer-Vietoris sequences for and , the short exact sequence

 0→Ω∗(X)→Ω∗(X1)⊕Ω∗(X2)→Ω∗(X0)→0,

the homotopy formula, and the exact sequences (2.1). ∎

2.12 Theorem.

If, in addition to the assumption of Theorem 2.10, is finitely generated for each and the torsion subgroup then there is an isomorphism of cohomology theories .

Proof.

We claim this statement as [27, Theorem 7.12]. However, the proof given there is not correct, and the assertion of [27, Theorem 7.12] unfortunately is slightly stronger than the one we can actually prove, namely Theorem 2.12.

As by Theorem 2.10 is a generalized cohomology theory, it is represented by a spectrum , and the natural transformation by a map of spectra. We extend this to a fiber sequence of spectra , inducing for each compact CW-complex an associated long exact sequence

 →F∗(X)→ER∗(X)a→^E∗flat(X)→. (2.12)

Comparison with (2.10) implies that the image of in coincides with the image of . This means by definition that the composed map of spectra is a phantom map. However, under the assumption that the is finitely generated for each , it is shown in [27, Section 8] that such a phantom map is automatically trivial. Using the triangulated structure of the homotopy category of spectra, we can choose , to obtain a map of fiber sequences (distinguished triangles)

 F−−−−→ERa−−−−→U⏐⏐↓ϕF⏐⏐↓=⏐⏐↓ϕE−−−−→ER−−−−→ER/Z.

with associated diagram of exact sequences which because of the knowledge about image and kernel of specializes to

 E∗(X)−−−−→ER∗(X)a−−−−→^E∗+1flat(X)−−−−→E∗+1tors(X)→0⏐⏐↓=⏐⏐↓=⏐⏐↓ϕE∗(X)−−−−→ER∗(X)−−−−→ER/Z∗(X)−−−−→E∗+1tors(X)→0. (2.12)

Note that we do not claim (and don’t know) whether the diagram can be completed to a commutative diagram by .

If , the -lemma implies that induces an isomorphism on the point and therefore for all finite CW-complexes. ∎

3 Uniqueness of differential extensions

Given the many different models of differential extensions of K-theory, many of which we are going to described in Section 4, it is reassuring that the resulting theory is uniquely determined. The corresponding statement for differential extensions of ordinary cohomology has been established in [62] by Simons and Sullivan. For K-theory and many other generalized cohomology theories we will establish this in the current section.

As in Subsection 2.3 we consider universal differential extensions of . Given two extensions and of with corresponding natural transformations as in 2.1, we are looking for a natural isomorphism compatible with the natural transformations. Provided such a natural transformation exists, we ask whether it is unique. We have seen that it often is natural to have additionally a transformation “integration over ” as in Definition 2.8, and we require that is compatible with this transformation, as well.

To construct the transformation in degree there is the following basic strategy:

• find a classifying space for with universal element . This means that for any space and , there is a map (unique up to homotopy) such that .

• lift this universal element to and show that this class is universal for , or at least that for a class one can find such that the difference is under good control.

• Obtain similarly . Define the transformation by and extend by naturality.

• Check that has all desired properties.

• Uniqueness of does follow if the lifts are uniquely determined by once their curvature is also fixed.

There are a couple of obvious difficulties implementing this strategy. The first is the fact, that the classifying space almost never has the homotopy type of a finite dimensional manifold. Therefore, is not defined. This is solved by replacing by a sequence of manifolds approximating with a compatible sequence of classes in which replace . Then, the construction of as indicated indeed is possible. However, a priori this has a big flaw. is not necessarily a transformation of abelian groups. Because of the compatibilities with , and the deviation from being additive is rather restricted and in the end is a class in . The different possible transformation are by naturality and the compatibility conditions determined by the different lift with fixed curvature. This indeterminacy is given by an element in . If is even and is rationally even, then so is as classifying space of . It then follows that and , i.e. automatically is additive and unique.

For odd, the transformation can then be defined (and is uniquely determined) by the requirement that it is compatible with . This construction has been carried out in detail in [27] and we arrive at the following theorem.

3.1 Theorem.

Assume that and are two differential extensions with -integration of a generalized cohomology theory which is rationally even, i.e.  for all . Assume furthermore that is a finitely generated abelian group for each . Then there is a unique natural isomorphism between these differential extensions compatible with the -integrations.

If no -orientation is given, the natural isomorphism can still be constructed on the even degree part.

If and are multiplicative, the transformation is automatically multiplicative. Note that the assumptions imply by Theorem 2.9 that then there is a canonical integration.

If and are defined on all manifold, not only on compact manifolds (possibly with boundary), then it suffices to require that is countably generated for each , and the same assertions hold.

Proof.

The proof is given in [27] and we don’t plan to repeat it here. However, there we made the slightly stronger assumption that if is only defined on the category of compact manifolds. Let us therefore indicate why the stronger result also holds. As described in the strategy, the first task is to approximate the classifying spaces for by spaces on which we can evaluate . These approximations are constructed inductively by attaching handles to obtain the correct homotopy groups (introducing new homotopy, but also killing superfluous homotopy). To construct a compact manifold, we are only allowed to attach finitely many handles. Therefore, we have to know a priori that we have to kill only a finitely generated homotopy groups. In [27] we assume that is zero, to be allowed to start with a simply connected approximation. Then we use that all homotopy groups of a simply connected finite CW-space are finitely generated. However, exactly the same holds for finite CW-spaces with finite fundamental group, because the higher homotopy groups are the homotopy groups of the universal covering, which in this case is a finite simply connected CW-complex. Note that a finitely generated abelian group with is automatically finite. ∎

3.2 Remark.

The general strategy leading toward Theorem 3.1 has been developed by Moritz Wiethaup 2006/07. However, his work has not been published yet. This was then taken up and developed further in [27].

3.1 Uniqueness of differential K-theory

We observe that all the assumptions of Theorem 3.1 are satisfied by K-theory, and also by real K-theory. Therefore, we have the following theorem:

3.3 Theorem.

Given two differential extensions and of complex K-theory, there is a unique natural isomorphism compatible with all the structure. If the extensions are multiplicative, this transformation is compatible with the products.

If both extensions come with -integration as in Definition 2.8 there is a unique natural isomorphism compatible with all the structure, including the integration.

In other words, all the different models for differential K-theory of Section 4 define the same groups —up to a canonical isomorphism.

3.4 Remark.

In Theorem 3.3 we really have to require the existence of -integration. In [27, Theorem 6.2] an infinite family of “exotic” differential extensions of K-theory are constructed. Essentially, the abelian group structure is modified in a subtle way in these examples to produce non-isomorphic functors which all satisfy the axioms of Section 2.

4 Models for differential K-theory

4.1 Vector bundles with connection

The most obvious attempt to construct differential cohomology (at least for ) is to use vector bundles with connection. It is technically convenient to also add an odd differential form to the cycles. This, indeed is the classical picture already used by Karoubi in [46, Section 7] for his definition of “multiplicative K-theory”, which we would call the flat part of differential K-theory.

4.1 Definition.

A cycle for vector bundle K-theory is a triple , where is a smooth complex Hermitean vector bundle over , a Hermitean connection on and a class of a differential form of odd degree. The curvature of a cycle essentially is defined as the Chern-Weil representative of the Chern character of , computed using the connection :

 R(E,∇,ω):=ch(∇)−dω.
4.2 Remark.

We require the use of Hermitean connections to obtain real valued curvature forms. Alternatively, one would have to use as target of cohomology with complex instead of cohomology with real coefficients. A slightly more extensive discussion of this matter can be found in [54, Section 2].

We define in the obvious way the sum . Two cycles and are equivalent if there is a third bundle with connection and an isomorphism such that

 ˜ch(∇⊕∇F,Φ−1(∇′⊕∇F)Φ)=ω−ω′,

where denotes the transgression Chern form between the two connections such that .

4.3 Remark.

In [39, Section 9], a model for differential is given where the cycles are Hermitean vector bundles with connection and a unitary automorphism, and an additional form (modulo the image of ) of even degree. The relations include in particular a rule for the composition of the unitary automorphisms: if and are two unitary automorphisms of then , where is the Chern-Simons form relating , and . We do not discuss this model in detail here.

4.2 Classifying maps

Hopkins and Singer, in their ground breaking paper [44] give a cocycle model of a differential extension of any cohomology theory (with transformation) , based on classifying maps. Here, for the construction of one has to choose two fundamental ingredients:

1. a classifying space for ; note that no smooth structure for is required (and could be expected)

2. a cocycle representing , so that, whenever represents a class , then represents . We can think of this cocycle as an -valued singular cocycle, although variations are possible.

A Hopkins-Singer cycle for then is a so called differential function, which by definition is a triple consisting of a continuous map , a closed differential -form with values in and an -cochain satisfying

 δh=ω−f∗c.

In other words: is an explicit representative for a class , and we are of course setting . is a de Rham representative of , and we are setting . This data actually gives two explicit representatives for , namely and (here, we have to map both and to a common cocycle model for like smooth singular cochains. defines such a cocycle by the de Rham homomorphism “integrate over the chain”, by restriction).

By definition, and are equivalent if and there is on which restricts to the two cycles on and .

The advantage of this approach is its complete generality. A disadvantage is that the cycles don’t have a nice geometric interpretation. Moreover, operations (like the addition and multiplication) rely on the choice of corresponding maps between the classifying spaces realizing those. These maps have then to be used to define the same operations on the differential cohomology groups. This is typically not very explicit. Moreover, properties like associativity, commutativity etc. will not hold on the nose for these classifying maps but are implemented by homotopies which have to be taken into account when establishing the same properties for the generalized differential cohomology. This can quickly get quite cumbersome and we refrain from carrying this out in any detail.

4.3 Geometric families of elliptic operators

In [26], a cycle model for differential K-theory (there called “smooth K-theory”) is developed which is based on local index theory. In spirit, it is similar to the passage of the classical model of K-theory via vector bundles to the Kasparov KK-model, where all families of generalized index problems are cycles.

Similarly, the cycles of [26] are geometric families of Dirac operators. It is clear that a lot of differential structure has to be present to obtain differential K-theory, so the definition has to be more restrictive than in Kasparov’s model. There are many advantages of an approach with very general cycles:

• first and most obvious, it is very easy to construct elements of differential K-theory if one has a broad class of cycles.

• the approach allows for a unified treatment of even and odd degrees

• the flexibilities of the cycles allows for explicit constructions in many contexts. In particular, it is easy to explicitly define the product and also the push-forward along a fiber-bundle.

It might seems as a disadvantage that one necessarily has a broad equivalence relation and that it is therefore hard to construct homomorphisms out of differential K-theory. To do this, one has to use the full force of local index theory. However, this is very well developed and one can make use of many properties as black box and then efficiently carry out the relevant constructions.

4.4 Definition.

Let be a compact manifold, possibly with boundary. A cycle for a is a pair , where is a geometric family, and is a class of differential forms.

A geometric family over (introduced in [19]) consists of the following data:

1. a proper submersion with closed fibers ,

2. a vertical Riemannian metric , i.e. a metric on the vertical bundle , defined as .

3. a horizontal distribution , i.e. a bundle such that .

4. a family of Dirac bundles ,

5. an orientation of .

Here, a family of Dirac bundles consists of

1. a Hermitean vector bundle with connection on ,

2. a Clifford multiplication ,

3. on the components where has even dimension a -grading .

We require that the restrictions of the family Dirac bundles to the fibers , , give Dirac bundles in the usual sense (see [19, Def. 3.1]):

1. The vertical metric induces the Riemannian structure on ,

2. The Clifford multiplication turns into a Clifford module (see [11, Def.3.32]) which is graded if is even.

3. The restriction of the connection to is a Clifford connection (see [11, Def.3.39]).

A geometric family is called even or odd, if is even-dimensional or odd-dimensional, respectively, and the form has the corresponding opposite parity.

4.5 Example.

The cycles for of Definition 4.1 are special cases of the cycles of Definition 4.4: is just , i.e. the fibers consist of the -dimensional manifold .

There are obvious notions of isomorphism (preserving all the structure) and of direct sum of cycles. We now introduce the structure maps and and the equivalence relation on the semigroup of isomorphism classes.

4.6 Definition.

The opposite of a geometric family is obtained by reversing the signs of the Clifford multiplication and the grading (in the even case) of the underlying family of Clifford bundles, and of the orientation of the vertical bundle.

4.7 Definition.

The usual construction of Dirac type operators of a Clifford bundle (compare [53, 11]), applied fiberwise, assigns to a geometric family over a family of Dirac type operators parameterized by , and this is indeed the main idea behind the geometric families. Then, the classical construction of Atiyah-Singer assigns to this family its (analytic) index , where is equal to the parity of the dimension of the fibers. In the special case of Example 4.5 —a vector bundle with connection over — this is exactly the K-theory class of the underlying -graded vector bundle.

We define .

4.8 Remark.

We define in such a way that . Moreover, is additive under sums of geometric families. The equivalence relation we are going to define will be compatible with .

We now proceed toward the definition of . It is based on the notion of local index form, an explicit de Rham representative of . It is one of the important points of the data collected in a geometric family that such a representative can be constructed canonically. For a detailed definition we refer to [19, Def. 4.8], but we briefly formulate its construction as follows. The vertical metric and the horizontal distribution together induce a connection on . Locally on we can assume that has a spin structure. We let be the associated spinor bundle. Then we can write the family of Dirac bundles as for a twisting bundle with metric, metric connection, and -grading which is determined uniquely up to isomorphism. The form is globally defined, and we get the local index form by applying the integration over the fiber :

 Ω(E):=∫E/B^A(∇Tvπ)∧ch(∇W) . (4.8)

The characteristic class version of the index theorem for families is

4.9 Theorem ([4]).

.

A proof using methods of local index theory has been given in [12], compare [11].

The equivalence relation we impose is based on the following idea: a geometric family with index should potentially be equivalent to the cycle , but in general only up to some differential form (with degree of shifted parity). Moreover, in this case the local index form will be exact, but it is important to find an explicit primitive with . Therefore, we identify geometric reasons why the index is zero and which provide such a primitive.

4.10 Definition.

A pre-taming of is a family of self-adjoint operators given by a smooth fiberwise integral kernel . In the even case we assume in addition that is odd with respect to the grading. The pre-taming is called a taming if is invertible for all . In this case, by definition is zero. However, as the index is unchanged by the smoothing perturbation , also if admits a taming.

Let be the notation for geometric family with a chosen taming. In [19, Def. 4.16], the -form is defined. By [19, Theorem 4.13]) it satisfies

 dη(Et)=Ω(E) . (4.10)

We skip the considerable analytic difficulties in the construction of the eta-form and use it and its properties as a black box.

4.11 Definition.

We call two cycles and paired if there exists a taming such that

 ρ−ρ′=η((E⊔BE′op)t) .

We let denote the equivalence relation generated by the relation ”paired”.

We define the differential -theory of to be the group completion of the abelian semigroup of equivalence classes of cycles as in Definition 4.4 with fiber dimension congruent modulo .

4.12 Theorem.

Definition 4.11 of indeed defines (with the obvious notion of pullback) a contravariant functor on the category of smooth manifolds which is a differential extension of K-theory.

The necessary natural transformations are induced by

1. of Definition 4.7

2.  of (4.8).

Proof.

This is carried out in [26, Section 2.4]. ∎

4.13 Definition.

Given two geometric families over a base , there is an obvious geometric way to define their fiber product , with underlying fiber bundle the fiber product of bundles of manifolds etc. Details are spelled out in [26, Section 4.1]. We then define the product of two cycles and (homogeneous of degree and , respectively) as

 (E,ρ)∪(F,θ