much of the material below has been or is being reworked into the entries Smooth∞Grpd and connection on a smooth principal ∞-bundle
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
In every cohesive (∞,1)-topos there is an intrinsic notion of Chern-Weil theory. We discuss the concrete realization of this in the cohesive -topos Smooth∞Grpd of smooth ∞-groupoids. This s the case that subsumes ordinary Chern-Weil theory of smooth principal bundles with connection and generalizes it to connections on smooth principal ∞-bundles.
The central motivation for the study of a higher generalization of ordinary Chern-Weil theory is the interest in extending the Chern-Weil homomorphism for a given Lie group to the higher connected covers of through the whole Whitehead tower of . Beyond the simply connected cover, these higher connected covers are still topological groups but fail to be (finite dimensional) Lie groups. They do however have natural realizations as smooth ∞-groups. Higher Chern-Weil theory is the extension of Chern-Weil theory from Lie groups to such smooth -groups. It allows the refinement of differential characteristic classes to fractional differential characteristic classes, that capture finer cohomological information.
Fractional differential classes
We give some examples of such fractional characteristic classes that occur in practice.
It is a familiar classical fact that the first Pontryagin class
which represents the generator of the fourth integral cohomology of the classifying space of the special orthogonal group allows a division by 2 when pulled back one step along the Whitehead tower to the classifying space of the spin group, in that there is a commuting diagram
in Top, where the top horizontal morphism represents a generator of the 4th integral cohomology of the classifying space of the spin group and the right vertical morphism is induced by multiplication by 2 on the additive group of integers.
This means that for manifold with spin structure exhibited by a classifying map
of its tangent bundle , the characteristic class of regarded as an -associated bundle contains less information than the class . For instance if the 4th cohomology of happens to be 2-torsion, the former class entirely vanishes, while the latter need not.
This familiar situation poses no problem to classical Chern-Weil theory, because both the special orthogonal group as well as the spin group of course have canonical structures of Lie groups, so that the Chern-Weil homomorphism may be applied to either. We shall write for the smooth refinement of the classifying space : the delooping Lie groupoid of or equivalently the moduli stack for smooth -principal bundles. Here and in the following the boldface indicates smooth (or otherwise cohesive) refinements. Accordingly, there is a smooth refinement of the first Pontryagin class, which takes smooth -principal bundles to their first Pontryagin class. This in turn has has a further differential refinement that takes -principal bundles with connection to their Chern-Simons 2-gerbes with connection.
All this is still captured by the traditional (refined) Chern-Weil homomorphism. But this is no longer the case as we keep climbing up the Whitehead tower of the orthogonal group. In the next step the second Pontryagin class may be divided by 6 when pulled back to the classifying space of the string group (SSSII)
As before, this means that if a space admits a string structure, then the characteristic class contains less information than the fractional refinement that it admits. In particular, the former may vanish if the degree 8 cohomology group of has 6-torsion, while the latter need not vanish.
For purposes of ordinary cohomology this is no problem, but for the differential refinement by ordinary Chern-Weil theory it is: the string group does not admit a Lie group structure that would make it a smooth version of the homotopy fiber of and hence standard Chern-Weil theory cannot produce the differential refinement of the fractional class .
But -Chern-Weil theory can: there is a natural smooth refinement of the string group to a Lie 2-group: the string 2-group. We write for the corresponding delooping ∞-Lie groupoid. The fractional second Pontryagin class does lift to this smooth refinement to produce a characteristic class
internal to ∞LieGrpd. Since this now lives in a smooth context, it does now have a differential Chern-Weil refinement
that takes smooth -principal 2-bundles with 2-connection to degree 8-cocycles in ordinary differential cohomology.
This kind of refinement we discuss in a bit more detail in the next section.
Higher differential spin structures
These refined differential invariants of fractional characteristic classes are relevant in the discussion of higher differential spin structures. (See the first part of (SatiSchreiberStasheff II for a review).) Ordinary spin structures on a manifold may be understood as trivializations of what are called quantum anomaly Pfaffian line bundles on the configuration space of the spinning quantum particle propagating on that manifold. (This physical origin is after all the origin of the term spin structure .) When these point-like super-particles are generalized to higher-dimensional -branes, the trivialization of the corresponding Pfaffian line bundles correspond to string structures for (this goes back to (Killingback) and (Witten) and has been made rigorous in (Bunke) then to fivebrane structures for (SatiSchreiberStasheff II)).
More precisely, the Pfaffian line bundles appearing here come equipped with a connection, and what matters is a trivialization of these bundles as bundles with connection. This refinement translates to differential refinements of the string structures and the fivebrane structures on . The differential form data of a twisted differential string structure constitutes what in the physics literature is called the Green-Schwarz mechanism. While this still can and has been captured with tools of ordinary Chern-Weil theory and ordinary differential cohomology (Freed, Waldorf) it has a natural formulation in higher Chern-Weil theory. Going beyond that, the magnetic dual Green-Schwarz mechanism can be seen to encode a twisted differential fivebrane structure and this is not practical to be studied without some higher geometry.
The following restates this in a bit more technical detail.
For the spin group, the first nontrivial characteristic class is the first fractional Pontryagin class given by a cocycle in ordinary integral cohomology . This induces a map
from isomorphism classes of topological -principal bundles to degree 4 integral cohomology.
If we assume that is a smooth manifold then we may consider the set
of isomorphism-classes of smooth -principal bundles. Here and in all of the following, the boldface in ”” indicates a refinement, here of the bare classifying space to a smooth incarnation.
Then ordinary Chern-Weil theory provides a refinement of the fractional Pontryagin class to a map to ordinary differential cohomology
The first point of passing to a higher category theory-refinement of this situation is that it allows to refine, in turn, these morphisms of cohomology sets to morphisms
of cocycle ∞-groupoids: here is the groupoid whose objects are smooth -principal bundles, and whose morphisms are smooth homomorphisms between these. Similarly denotes the [[3-groupoid] whose objects are smooth circle 2-group-principal 3-bundles, while is accordingly the 3-groupoid whose objects are circle 3-bundles with connection, whose morphisms are homomorphisms between these, whose 2-morphisms are higher homotopies between those. The original morphism of cohomology sets is the decategorification of this, the restriction to connected components:
This refinement to cocylce -groupoids notably has the consequence that it allows us to produce the homotopy fibers of these morphisms. To see the relevance of this, recall (from string structure ) that the homotopy fiber of the bare fractional Pontryagin class, which is the (∞,1)-pullback/homotopy pullback
defines the -groupoid of string structures on ( smooth , but not differential ).
We can now replace the class by its differential refinement and obtain an ∞-groupoid that differentially refines the 2-groupoid of String-structures as the (∞,1)-pullback
This we may call the -groupoid of differential string-structures . A cocycle in there is naturally identified with a tuple consisting of
We may think of this as a refinement of secondary characteristic classes: the first Pontryagin curvature characteristic form itself is constrained to vanish, and so the Chern-Simons form 3-connection itself constitutes cohomological data.
So far this uses mostly just a little bit of (∞,1)-category theory or at least some homotopy theory. The first glimpse of something beyond ordinary Chern-Weil theory appearing is the -groupoid which may be thought of as the 2-groupoid of smooth string 2-group-principal 2-bundles.
But suppose we fix an such that is nontrivial. Then we can continue the proceed to higher degrees:
the next topological characteristic class is the second fractional Pontryagin class . Since the string group does not have the structure of a Lie group, this cannot be refined to differential cohomology using ordinary Chern-Weil theory. However, in terms of -Chern-Weil theory it can:
we may obtain a differential refinement
that maps smooth string 2-group-principal 2-bundles with 2-connectins to their Chern-Simons circle 7-bundle with connection. This is an example of the higher version of the Chern-Weil homomorphism.
And naturally we are then entitled to form its homotopy fibers and produce the 7-groupoid of differential fivebrane structures – . For that notice (see fivebrane structure) that the homotopy fiber of the smooth but non-differential cocycles
is the 7-groupoid of smooth fivebrane structures on . Its differential refinement
we may therefore call the 7-groupoid of differential fivebrane structures . Cocycles in here are naturally identified with tuples of
These are the kind of structures that -Chern-Weil theory studies.
Ordinary Chern-Weil theory is about refinements of characteristic classes of -principal bundles for a Lie group (equivalently of the classifying space of that Lie group) from ordinary cohomology to differential cohomology.
Under -Chern-Weil theory we want to understand the generalization of this to (∞,1)-category theory: where Lie groups are generalized to ∞-Lie groups, Lie algebras are generalized to ∞-Lie algebras and principal bundles to principal ∞-bundles.
So -Chern-Weil theory produces differential cohomology-refinements of characteristic classes of -principal ∞-bundles for an ∞-Lie group, equivalently of the corresponding classifying spaces .
Ordinary Chern-Weil theory is formulated in the context of differential geometry. We need to widen this context somewhat in order that it can accomodate the relevant higher structures and so we place ourselves in the context of the (∞,1)-topos ∞LieGrpd of ∞-Lie groupoids.
In every -topos that admits a notion of differential cohomology, there is a general abstract notion of refinement of characteristic classes in cohomology to curvature characteristic classes in ordinary differential cohomology.
The main construction in ∞-Chern-Weil theory is a concrete model or presentation of this abstract operation. This model is constructed in terms of Lie integration of objects in ∞-Lie algebra cohomology. This construction is the higher analog of the Chern-Weil homomorphism. Its crucial intermediate step is the definition and construction of ∞-connections on principal ∞-bundles.
This model itself is after all built on concrete familiar constructions in differential geometry and can be studied and appreciated in itself without recourse to the higher topos theory that we claim it provides a model for. The so inclined reader can ignore all the general abstract discussion in the following and concentrate on the concrete differential geometry.
ere is how this entry here proceeds.
A warmup for the full theory that connects to classical constructions is given at
we discuss the general definition of -connections and of the Chern-Weil homomorphism and discuss some general properties. Then we turn to discussing
General -Chern-Weil theory, as described below, is naturally formulated in the context of (infinity,1)-topos-theory and some of its aspects can only be understood from that perspective.
However, unwinding the abstract higher topos theoretic concepts in terms of 1-categoriecal models yields concrete structures in familiar contexts of differential geometry that connect to various classical and familiar concepts. Since a full appreciation of the abstract formulation benefits from having a feeling for how these concrete models work out, the reader may at this point wish to look into some such basic aspects. These may be found behind the following link
For ∞-groups in an ∞-connected (∞,1)-topos with deloopings and , respectively, every characteristic class serves to pull back the canonical intrinsic curvature form to an intrinsic differential form on .
For an ordinary Lie group regarded naturally as an object in ∞LieGrpd, we show that the ordinary Chern-Weil homomorphism for -principal bundles may be understood as a concrete model for this simple abstract situation, which applies to those characteristic classes that happen to be in the image of the Lie intgeration of Lie algebra cocycles.
More generally, this construction applies for an ∞-Lie group with ∞-Lie algebra and a characteristic class on that arises from Lie integration of a cocycle in the ∞-Lie algebra cohomology of .
The ordinary Chern-Weil homomorphism uses a connection on a bundle as an intermediate tool for interpolating from a -principal bundle to its curvature characteristic, represented by the curvature characteristic form , where is the curvature of and is an invariant polynomial on . The choice of connection in this construction may be understood as providing a correspondence space in the following construction.
We know from the discussion of abelian differential cohomology above that the intrinsic morphism
in is modeled in by the correspondence
If we write
and so forth, then this correspondence is
If now is modeled by the Lie integration of a cocycle on a Lie -algebra
for , then the total intrinsic differential form is modeled by the zig-zag of morphisms
in . In order to compute with such zig-zags of morphisms, in particular in order to compute homotopy fibers, it is helpful to complete this to a single correspondence. There is a fairly evident choice for the tip of this total corresponence, namely
It remains to complete the square and extend the ∞-Lie algebra cocycle to a morphism . This is accomplished by an invariant polynomial
which is in transgression with , witnessed by the Chern-Simons element . Using this, we obtain the total diagram
By the fact that this commutes, we have that the correspondence
in models the intrinsic curvature characteristic form .
We may identify cocycles with values in as (pseudo)--connections on the underlying -cocycle. If their curvature is represented by a cocycle in which is given by a globally defined form, then these are genuine -connections. In either case, they serve as an intermediate step in computing the curvature characteristics.
-Lie algebra valued connections
The content of this section is at connection on an infinity-bundle.
(Chern-Weil curvature characteristics)
Let be an invariant polynomial on the Lie n-algebra . Postcomposition with the corresponding diagram of dg-algebras
induces a morphism of simplicial presheaves
into the -coskeleton of the model for the de Rham coefficient object discussed above.
For a connection, we call the induced intrinsic de Rham cocycle
the Chern-Weil curvature characteristic form of with respect to .
For a genuine connection, the induced curvature characteristic forms are globally defined closed forms, in that their cocycle factors through the sheaf of closed -forms:
for given notice that is closed and for a genuine connection has no leg along : for a vector field along we have . Therefore the Lie derivative along a vector along the simplex vanishes:
So a genuine -Lie algebra valued connection is a cocycle with values in the -coskeleton of the simplicial presheaf of diagrams, which over assigns the set of diagrams
(with the dg-algebra of vertical differential forms on the bundle ), where the top morphism encodes the cocycle for the underlying -principal ∞-bundle, where the middle morphism encodes the connection data and the bottom morphism the curvature characteristic forms.
Such -Lie algebra valued connections were introduced in SSSI and further studied in SSSIII.
Higher order Chern-Simons forms
See at Chern-Simons form the section In ∞-Chern-Weil theory.
Above we have considered ∞-Lie algebra valued connections and their curvature characteristic forms. We now wish to show how these model the intrinsic Chern character in an (∞,1)-topos.
Since our ambient (∞,1)-topos is assumed to be locally ∞-connected we have in addition to the notion of Postnikov tower in an (∞,1)-category the notion of Whitehead tower in an (∞,1)-topos. Both notions are dual to each other: for any object and
the intrinsic Postnikov tower of its path ∞-groupoid, the pasting composite of (∞,1)-pullbacks
defines the Whitehead tower
Since our is assumed to be even ∞-connected, the Postnikov tower of is the image under of the ordinary Postnikov tower of in . Accordingly, we have
The point now is that in ∞LieGrpd we may form smooh refinements of these discrete extensions: every discrete -group we want to refine to a smooth -group . By the discussion at geometric realization, both have equivalent underlying -groupoids
For every direct summand abelian group in one of the we can ask for a refinement of the cocycle from coefficients to . This does not change the geometric realization, up to equivalence, but does change the smooth structure. And it allows to refine to differential coefficients by postcomposing further with .
For instance for the delooping of the spin group, we refine the internal Whitehead tower to
where the deloopings of the string 2-group and the fivebrane 6-group appear.
The result of such a smooth refinement is that we may apply the intrinsic curvature classes and the intrinsic de Rham theorem to obtain cocycles in realified cohomology, for instance
If we have a -principal bundle , we may form over it the covering circle -group bundles on which these higher cocycles naturally live
Here for an ordinary space, , the higher circle -group principal bundes have the property that also . Therefore the 0-truncation of the entire composite
defines a closed 8-form on . This is the curvature characeristic form given by the Chern-Weil homomorphism in this degree. Its refinement to Deligne cohomology in this construction lives naturally not on , but on the covering of .
Principal 1-bundles with connection
We spell out here how the general theory of ∞-Lie algebra valued connection reduces to the standard notion of connections on ordinary -principal bundles and how the ∞-Chern-Weil homomorphism reduces on these to the ordinary Chern-Weil homomorphism.
Let be a (finite dimensional) Lie algebra. Then
is the delooping of the simply connected Lie group integrating .
The coefficient object of genuine ∞-Lie algebra connections for an ordinary Lie algebra is weakly equivalent to the simplicial presheaf
that assigns objectwise the groupoid of Lie-algebra valued 1-forms.
This is moreover isomorphic to the simplicial presheaf
of morphisms out of the path groupoid.
The flat coefficient object is modeled by the subobject
of groupoids of Lie-algebra valued forms with vanishing curvature 2-form.
This is isomorphic to
of morphism out of the fundamental groupoid.
The statements about morphisms out of the path groupoid are discussed in detail in SchrWalI.
For a paracompact smooth manifold and a good open cover we have a natural equivalence of groupoids
with the groupoid of smooth -principal bundles with connection on .
For an invariant polynomial on , the induced morphism
is that of the ordinary Chern-Weil homomorphism.
We have seen that a refinement of the Chern-Weil homomorphism is available. The above morphism extends to a morphism
in represented by
For a smooth manifold with good cover we have that is the groupoid whose objects are closed -forms on and whose morphisms are given by -forms modulo exact forms.
Let range over a set of generators for all invariant polynomials. Then
is an approximation to the intrinsic Chern-character. We may consider its homotopy fibers over a given set of curvature characteristic forms.
Assume are two genuine connections with coinciding curvature characteristic classes . Then in the homotopy fiber they are coboundant cocycles precisely if all the Chern-Simons forms vanish modulo an exact form.
This equivalence relation is that which defines Simons-Sullivan structured bundles. Their Grothendieck group completion yields differential K-theory.
Principal 2-bundles with connection
Let be a Lie strict 2-group coming from a differential crossed module . Then we have two candidate Lie 2-groups integrating this: on the one hand the strict 2-group coming from the crossed module that integrates degreewise as ordinary Lie algebras, and on the other hand .
given by evaluating 2-dimensional parallel transport is a weak equivalence.
Use the 3-dimensional nonabelian Stokes theorem from the appendix of SchrWalII.
This is described in detail in SchrWalII, subject to the extra constraint that the 2-form curvature vanishes.
A genuine connection on a -principal 2-bundle with given cocycle is a cocycle given as follows:
- on a pair of forms , ;
1 on a function and a 1-form such that
- and so forth
This is described in detail in SchrWalIII, subject to the extra constraint that the 2-form curvature vanishes.
Twisted differential and -structures
We discuss now in detail refined Chern-Weil morphisms
that send ∞-connections on -principal ∞-bundles to circle n-bundles with connection that represent a given characteristic class. with coefficients in the circle n-groupoid.
Specifically, we consider the first two steps in the smooth refinement of the Whitehead tower of the orthogonal group that are controled by ∞-Lie algebra cohomology.
The smooth Whitehead tower of in ∞LieGrpd starts as
the delooping of the special orthogonal group is the -principal bundle over classified by the cocycle that sends an elemen to if it is in the connected component of the identity and to if it is not. This means we have an (∞,1)-pullback diagram
the delooping of the spin group is the is the -principal 2-bundle over classified by the Stiefel-Whitney class
Since these two steps are controled by the torsion-group they have no nontrivial refinement to differential cohomology. The next step however is controled by what in the (∞,1)-topos ∞Grpd Top is the first fractional Pontryagin class? and which lifts through the path ∞-groupoid functor to a characteristic class in ∞LieGrpd (as discussed there) with coefficients in the smooth circle 3-groupoid. This cocycle does arise as the Lie integration of the canonical Lie algebra 3-cocycle .
The principal 3-bundle that this classifies is the delooping of the string 2-group
Notice that the fact that this is an (∞,1)-pullback implies that for any also
which exhibits the 2-groupoid of string structures.
As we refine in this diagram the bottom morphism to differential cohomology, we obtain correspondingly differential string structures.
The string-lifting Chern–Simons -bundle with connection
We describe the special case of the general -Chern–Weil homomorphism for -Lie algebra valued connections corresponding to the characteristic class : the first fractional Pontryagin class of the spin group . The -differential cocycle that it produces from a given -principal bundle is the Chern–Simons 2-bundle with connection whose class is the obstruction for the existence of a string structure.
The content of this subsection is at Chern–Simons 2-gerbe in the section on -Chern–Weil theory.
Differential string structures
The content of this section is at differential string structure.
The Fivebrane-lifting Chern-Simons 7-bundle with connection
The content of this section is at Chern-Simons circle 7-bundle.
Differential fivebrane structures
be the differential refinement of the second fractional Pontryagin class discussed above.
For ∞LieGrpd, the -groupoid of differential fivebrane-structures is the homotopy fiber of .
More generally, the -groupoid of twisted differential fivebrane structures is the (∞,1)-pullback in
In terms of the underlying -Lie algebra valued local connection data, i.e. before Lie integration in the above sense , this has been considered in SSSIII
The refined higher Chern-Weil homomorphism takes values in circle n-bundles with connection in ordinary differential cohomology. Each of these comes with a notion of higher holonomy over -dimensional curves . The map that takes a connection on an infinity-bundle to this holonomy is a generalization of the action functional of Chern-Simons theory.
Therefore the higher Chern-Weil homomorphism defines a class of sigma-model quantum field theories that we call
See there for more details.
Special noteworthy cases are
An explicit presentation of the -Chern-Weil homomorphism in terms of simplicial presheaves and the application to differential string structures and differential fivebrane structures is considered in
The special case that gives the AKSZ sigma-model is discussed in
A general abstract account is in
For a commented list of related literature see
differential cohomology in cohesive topos – references