# nLab Chern-Weil theory in Smooth∞Grpd

much of the material below has been or is being reworked into the entries Smooth∞Grpd and connection on a smooth principal ∞-bundle

### Context

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

In every cohesive (∞,1)-topos there is an intrinsic notion of Chern-Weil theory. We discuss the concrete realization of this in the cohesive $(\infty,1)$-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.

## Motivation

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 $G$ to the higher connected covers of $G$ through the whole Whitehead tower of $G$. 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 $\infty$-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

$p_1 : \mathcal{B}SO \to \mathcal{B}^4 \mathbb{Z} \,,$

which represents the generator of the fourth integral cohomology of the classifying space $\mathcal{B} SO$ 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

$\array{ \mathcal{B} Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathcal{B}^4 \mathbb{Z} \\ \downarrow && \downarrow^{\mathrlap{\cdot 2}} \\ \mathcal{B} SO &\stackrel{p_1}{\to}& \mathcal{B}^4 \mathbb{Z} } \,,$

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 $X$ manifold with spin structure exhibited by a classifying map $\hat g$

$\array{ && \mathcal{B} Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathcal{B}^4 \mathbb{Z} \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow && \downarrow^{\mathrlap{\cdot 2}} \\ X &\stackrel{g}{\to}& \mathcal{B} SO &\stackrel{p_1}{\to}& \mathcal{B}^4\mathbb{Z} }$

of its tangent bundle $T X$, the characteristic class $p_1(T X) : X \stackrel{g}{\to} \mathcal{B}SO \stackrel{p_1}{\to} \mathcal{B}^4 \mathbb{Z}$ of $T X$ regarded as an $SO$-associated bundle contains less information than the class $\frac{1}{2}p_1(T X) : X \stackrel{\hat g}{\to} \mathcal{B}Spin \stackrel{\frac{1}{2}p_1}{\to} \mathcal{B}^4 \mathbb{Z}$. For instance if the 4th cohomology of $X$ 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 $\mathbf{B} \mathrm{Spin}$ for the smooth refinement of the classifying space $B \mathrm{Spin}$: the delooping Lie groupoid of $\mathrm{Spin}$ or equivalently the moduli stack for smooth $\mathrm{Spin}$-principal bundles. Here and in the following the boldface indicates smooth (or otherwise cohesive) refinements. Accordingly, there is a smooth refinement $\frac{1}{2}\mathbf{p} : \mathbf{B} \mathrm{Spin} \to \mathbf{B}^3 U(1)$ of the first Pontryagin class, which takes smooth $\mathrm{Spin}$-principal bundles to their first Pontryagin class. This in turn has has a further differential refinement $\frac{1}{2}{\hat {\mathbf{p}}} : \mathbf{B}\mathrm{Spin}_{\mathrm{conn}} \to \mathbf{B}^3 U(1)_{\mathrm{conn}}$ that takes $\mathrm{Spin}$-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 $p_2 : \mathcal{B}SO \to \mathcal{B}^8 \mathbb{Z}$ may be divided by 6 when pulled back to the classifying space of the string group (SSSII)

$\array{ \mathcal{B} String &\stackrel{\frac{1}{6}p_2}{\to}& \mathcal{B}^8 \mathbb{Z} \\ \downarrow && \downarrow^{\mathrlap{\cdot 6}} \\ \mathcal{B} SO &\stackrel{p_2}{\to}& \mathcal{B}^8 \mathbb{Z} } \,,$

As before, this means that if a space $X$ admits a string structure, then the characteristic class $p_2(X)$ contains less information than the fractional refinement $\frac{1}{6}p_2(X)$ that it admits. In particular, the former may vanish if the degree 8 cohomology group of $X$ 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 $\frac{1}{2}p_1$ and hence standard Chern-Weil theory cannot produce the differential refinement of the fractional class $\frac{1}{6}p_2$.

But $\infty$-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 $\mathbf{B}String$ for the corresponding delooping ∞-Lie groupoid. The fractional second Pontryagin class does lift to this smooth refinement to produce a characteristic class

$\frac{1}{6}\mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)$

internal to $\mathbf{H} =$ ∞LieGrpd. Since this now lives in a smooth context, it does now have a differential Chern-Weil refinement

$\frac{1}{6}\hat \mathbf{p}_2 : \mathbf{H}_{conn}(-, \mathbf{B}String) \to \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$

that takes smooth $String$-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 $p$-branes, the trivialization of the corresponding Pfaffian line bundles correspond to string structures for $p = 1$ (this goes back to (Killingback) and (Witten) and has been made rigorous in (Bunke) then to fivebrane structures for $p = 5$ (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 $X$. 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 $G = Spin$ the spin group, the first nontrivial characteristic class is the first fractional Pontryagin class given by a cocycle $\frac{1}{2}p_1 : \mathcal{B}G \to K(\mathbb{Z}, 4)$ in ordinary integral cohomology $H^4(\mathcal{B}Spin, \mathbb{Z})$. This induces a map

$H^1(X, Spin) = H(X, \mathcal{B}Spin) \to H^4(X, \mathbb{Z})$

from isomorphism classes of topological $Spin$-principal bundles to degree 4 integral cohomology.

If we assume that $X$ is a smooth manifold then we may consider the set

$Spin Bund(X)/ \sim = H(X,\mathbf{B}Spin)$

of isomorphism-classes of smooth $Spin$-principal bundles. Here and in all of the following, the boldface in ”$\mathbf{B}G$” indicates a refinement, here of the bare classifying space $\mathcal{B}G$ to a smooth incarnation.

Then ordinary Chern-Weil theory provides a refinement of the fractional Pontryagin class $H(X, \mathbf{B}Spin) \to H^4(X,\mathbb{Z})$ to a map to ordinary differential cohomology $H_{diff}^4(X)$

$\frac{1}{2} \hat p_1 : H(X, \mathbf{B}Spin) \to H_{diff}^4(X) \,.$

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

$\frac{1}{2} \mathbf{p}_1 : \mathbf{H}(X, \mathbf{B}Spin) \to \mathbf{H}(X,\mathbf{B}^3 U(1))$

and

$\frac{1}{2} \hat \mathbf{p}_1 : \mathbf{H}_{conn}(X, \mathbf{B}Spin) \to \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))$

of cocycle ∞-groupoids: here $\mathbf{H}(X,\mathbf{B}G)$ is the groupoid whose objects are smooth $Spin$-principal bundles, and whose morphisms are smooth homomorphisms between these. Similarly $\mathbf{H}(X,\mathbf{B}^3 U(1))$ denotes the [[3-groupoid] whose objects are smooth circle 2-group-principal 3-bundles, while $\mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))$ 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:

$\frac{1}{2}p_1 = \pi_0(\frac{1}{2}\mathbf{p}_1) \,.$

This refinement to cocylce $\infty$-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

$\array{ \mathbf{H}(X,\mathbf{B}String) &\to& * \\ \downarrow &\swArrow_\simeq& \downarrow \\ \mathbf{H}(X,\mathbf{B}G) &\stackrel{\frac{1}{2} \mathbf{p}_1}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) } \,,$

defines the $\infty$-groupoid $\mathbf{H}(X, \mathbf{B}String)$ of string structures on $X$ ( smooth , but not differential ).

We can now replace the class $\frac{1}{2}\mathbf{p}_1$ by its differential refinement $\frac{1}{2}\hat \mathbf{p}_1$ and obtain an ∞-groupoid $String_{diff}(X)$ that differentially refines the 2-groupoid $\mathbf{H}(X,\mathbf{B}String)$ of String-structures as the (∞,1)-pullback

$\array{ String_{diff}(X) &\to& * \\ \downarrow &\swArrow_\simeq& \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}G) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) } \,.$

This $String_{diff}(X)$ we may call the $\infty$-groupoid of differential string-structures . A cocycle in there is naturally identified with a tuple consisting of

• a smooth $Spin$-principal bundle $P \to X$ with connection $\nabla$;

• the Chern-Simons 2-gerbe with connection $CS(\nabla)$ induced by this;

• a choice of trivialization of this Chern-Simons 2-gerbe – this is the homotopy 2-morphism in the middle of the above pullback diagram.

We may think of this as a refinement of secondary characteristic classes: the first Pontryagin curvature characteristic form $\langle F_\nabla \wedge F_\nabla \rangle$ 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 $\infty$-groupoid $\mathbf{H}(X,\mathbf{B}String)$ which may be thought of as the 2-groupoid of smooth string 2-group-principal 2-bundles.

But suppose we fix an $X$ such that $H(X, \mathbf{B}String)$ is nontrivial. Then we can continue the proceed to higher degrees:

the next topological characteristic class is the second fractional Pontryagin class $\frac{1}{6}p_2 : \mathcal{B}String \to \mathcal{B}^7 U(1)$. 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 $\infty$-Chern-Weil theory it can:

we may obtain a differential refinement

$\frac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn}$

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$Fivebrane_{diff}(X)$. For that notice (see fivebrane structure) that the homotopy fiber of the smooth but non-differential cocycles

$\array{ \mathbf{H}(X, \mathbf{B}Fivebrane) &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{H}(X, \mathbf{B}String) &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{H}(X, \mathbf{B}^7 U(1)) }$

is the 7-groupoid of smooth fivebrane structures on $X$. Its differential refinement

$\array{ Fivebrane_{diff}(X) &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{H}_{conn}(X, \mathbf{B}String) &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^7 U(1)) }$

we may therefore call the 7-groupoid $Fivebrane_{diff}(X)$ of differential fivebrane structures . Cocycles in here are naturally identified with tuples of

• a $String$-principal 2-bundle $P \to X$, equipped with a 2-connection $\nabla$;

• the Chern-Simons circle 7-bundle $CS_7(\nabla)$ with connection induced by it;

• a choice of trivialization of $CS_7(\nabla)$.

These are the kind of structures that $\infty$-Chern-Weil theory studies.

## Idea

Ordinary Chern-Weil theory is about refinements of characteristic classes of $G$-principal bundles for $G$ a Lie group (equivalently of the classifying space $\mathcal{B}G$ of that Lie group) from ordinary cohomology to differential cohomology.

Under $\infty$-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 $\infty$-Chern-Weil theory produces differential cohomology-refinements of characteristic classes of $G$-principal ∞-bundles for $G$ an ∞-Lie group, equivalently of the corresponding classifying spaces $\mathcal{B}G$.

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 $\mathbf{H} =$ ∞LieGrpd of ∞-Lie groupoids.

In every $(\infty,1)$-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

Then in

we discuss the general definition of $\infty$-connections and of the Chern-Weil homomorphism and discuss some general properties. Then we turn to discussing

## Preparatory concepts

General $\infty$-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

## $\infty$-Chern-Weil theory

For $G,A$ ∞-groups in an ∞-connected (∞,1)-topos $\mathbf{H}$ with deloopings $\mathbf{B}G$ and $\mathbf{B}A$, respectively, every characteristic class $c : \mathbf{B}G \to A$ serves to pull back the canonical intrinsic curvature form $curv_A : A \to \mathbf{\flat}_{dR} \mathbf{B}A$ to an intrinsic differential form $curv_A\circ c : \mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}A$ on $\mathbf{B}G$.

For $G$ an ordinary Lie group regarded naturally as an object in $\mathbf{H} =$ ∞LieGrpd, we show that the ordinary Chern-Weil homomorphism for $G$-principal bundles may be understood as a concrete model for this simple abstract situation, which applies to those characteristic classes $c$ that happen to be in the image of the Lie intgeration of Lie algebra cocycles.

More generally, this construction applies for $G$ an ∞-Lie group with ∞-Lie algebra $\mathfrak{g}$ and $c$ a characteristic class on $\mathbf{B}G$ that arises from Lie integration of a cocycle in the ∞-Lie algebra cohomology of $\mathfrak{g}$.

The ordinary Chern-Weil homomorphism uses a connection on a bundle $\nabla$ as an intermediate tool for interpolating from a $G$-principal bundle to its curvature characteristic, represented by the curvature characteristic form $\langle F_\nabla \rangle$, where $F_\nabla$ is the curvature of $\nabla$ and $\langle - \rangle$ is an invariant polynomial on $\mathfrak{g}$. 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

$\mathbf{B}^n \mathbb{R}/\mathbb{Z} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}/\mathbb{Z}$

in $\mathbf{H} = \infty LieGrpd$ is modeled in $[CartSp^{op}, sSet]_{proj,cov}$ by the correspondence

$\array{ \mathbf{B}^n \mathbb{R}/\Gamma_{diff,simp} &\stackrel{curv}{\to}& \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}_{simp} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n \mathbb{R}/\Gamma_{simp} } \,.$

If we write

$\exp(b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R})) : \mathbf{cosk}_{n+1}( (U,[k]) \mapsto \left\{ \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^k) &\leftarrow& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(b^{n-1}\mathbb{R}) } \right\} )$

and so forth, then this correspondence is

$\array{ \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(b^{n-1}\mathbb{R}\to *) } \,.$

If now $\mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\mathbb{Z}$ is modeled by the Lie integration of a cocycle $\mu$ on a Lie $k$-algebra

$\mathbf{cosk}_{k+1} \exp(\mathfrak{g}) \stackrel{\exp(\mu)}{\to} \exp(b^{n-1}\mathbb{R})/\Gamma = \exp(b^{n-1}\mathbb{R} \to *)/\Gamma$

for $k \geq n-1$, then the total intrinsic differential form $\mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\Gamma \to \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}$ is modeled by the zig-zag of morphisms

$\array{ && \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{cosk}_{k+1}\exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to *) }$

in $[CartSp^{op}, sSet]$. 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

$\mathbf{B}G_{diff} := \mathbf{cosk}_{n+1} \exp(\mathfrak{g} \to inn(\mathfrak{g})) \,.$

It remains to complete the square and extend the ∞-Lie algebra cocycle $\mu : \mathfrak{g} \to b^{n-1}\mathb{R}$ to a morphism $(\mathfrak{g} \to inn(\mathfrak{g})) \to (b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R}))$. This is accomplished by an invariant polynomial

$\langle -\rangle_\mu : inn(\mathfrak{g}) \stackrel{(\langle - \rangle_\mu, cs_\mu)}{\to} inn(b^{n-1}\mathbb{R}) \to b^n \mathbb{R}$

which is in transgression with $\mu$, witnessed by the Chern-Simons element $cs_\mu$. Using this, we obtain the total diagram

$\array{ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to inn(\mathfrak{g})) &\stackrel{(\langle - \rangle_\mu, cs_\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{cosk}_{k+1}\exp(\mathrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to *) } \,.$

By the fact that this commutes, we have that the correspondence

$\left( \array{ \mathbf{B}G_{diff,simp} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{simp} \\ \downarrow \\ \mathbf{B}G_{simp} } \right) \;\; := \;\; \left( \array{ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to inn(\mathfrak{g})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow \\ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to *) } \right)$

in $[CartSp^{op}, sSet]_{proj,cov}$ models the intrinsic curvature characteristic form $\mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}$.

We may identify cocycles with values in $\mathbf{B}G_{diff}$ as (pseudo)-$\infty$-connections on the underlying $\mathbf{B}G$-cocycle. If their curvature is represented by a cocycle in $\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}$ which is given by a globally defined form, then these are genuine $\infty$-connections. In either case, they serve as an intermediate step in computing the curvature characteristics.

### $\infty$-Lie algebra valued connections

The content of this section is at connection on an infinity-bundle.

### Curvature characteristics

###### Definition

(Chern-Weil curvature characteristics)

Let $\langle -\rangle : inn(\mathfrak{g}) \to b^{p} \mathbb{R}$ be an invariant polynomial on the Lie n-algebra $\mathfrak{g}$. Postcomposition with the corresponding diagram of dg-algebras

$\array{ CE(\mathfrak{g}) &\leftarrow& 0 \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& CE(b^p \mathbb{R}) }$

induces a morphism of simplicial presheaves

$\langle F_{(-)} \rangle : \mathbf{B}G_{diff} \to \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1}\mathbb{R}_{simp}$

into the $(n+1)$-coskeleton of the model for the de Rham coefficient object $\mathbf{\flat}_{dR}\mathbf{B}^{p+1}\mathbb{R}$ discussed above.

For $\nabla : \hat X \to \mathbf{B}G_{diff}$ a connection, we call the induced intrinsic de Rham cocycle

$\langle F_\nabla \rangle : \hat X \stackrel{\nabla}{\to} \mathbf{B}G_{diff} \stackrel{\langle -\rangle}{\to} \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1}\mathbb{R}_{simp}$

the Chern-Weil curvature characteristic form of $\nabla$ with respect to $\langle -\rangle$.

###### Lemma

For $\nabla : \hat X \to \mathbf{B}G_{conn} \hookrightarrow \mathbf{B}G_{diff}$ a genuine connection, the induced curvature characteristic forms are globally defined closed forms, in that their cocycle factors through the sheaf $\Omega^{p+1}_{cl}(-)$ of closed $(p+1)$-forms:

$\array{ && \mathbf{B}G_{conn} &\stackrel{\langle F_{(-)}\rangle}{\to}& \Omega^{p+1}_{cl}(-) \\ & \nearrow & \downarrow && \downarrow \\ \hat X &\stackrel{}{\to}& \mathbf{B}G_{diff} &\stackrel{\langle F_{(-)}\rangle}{\to}& \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1} \mathbb{R}_{simp} } \,.$
###### Proof

for given $(U,[k])$ notice that $\langle F_{\nabla}\rangle(U,[k]) \in \Omega^\bullet(U\times \Delta^k)$ is closed and for $\nabla$ a genuine connection has no leg along $\Delta^k$: for $\partial_t$ a vector field along $\Delta^k$ we have $\iota_{\partial_t} \langle F_A\rangle = 0$. Therefore the Lie derivative along a vector $\partial_t$ along the simplex vanishes:

$\mathcal{L}_t \langle F_A\rangle = d \iota_t \langle F_A\rangle + \iota_t d \langle F_A\rangle = 0 \,.$
###### Remark

As for the groupal case above, we hence find that the genuine $\infty$-connections are selected among all pseudo-connections as those whose curvature characteristic has a 0-truncated cocycle representative.

So a genuine $\infty$-Lie algebra valued connection is a cocycle with values in the $(n+1)$-coskeleton of the simplicial presheaf of diagrams, which over $U,[k]$ assigns the set of diagrams

$\array{ C^\infty(U \times \Delta k)_{vert} &\leftarrow& CE(\mathfrak{g}) &&& cocycle\;for\;underlying\;G-principal\;\infty-bundle \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(X) &\leftarrow& CE(inn(\mathfrak{g})) = W(\mathfrak{g}) &&& connection\;and\;curvature \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\leftarrow& inv(\mathfrak{g}) &&& curvature\;characteristic\;forms } \,,$

(with $\Omega^\bullet(U \times \Delta^k)_{vert}$ the dg-algebra of vertical differential forms on the bundle $U \times \Delta^k \to U$), where the top morphism encodes the cocycle for the underlying $G = \tau_n\exp(\mathfrak{g})$-principal ∞-bundle, where the middle morphism encodes the connection data and the bottom morphism the curvature characteristic forms.

Such $\infty$-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.

### Chern character

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.

$ch_{\mathbf{B}G} : \mathbf{B}G \to \mathbf{\Pi}(\mathbf{B}G) \to \mathbf{\Pi}(\mathbf{B}G)\otimes R \,.$

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 $A \in \mathbf{H}$ any object and

$\mathbf{\Pi}(A) \to \cdots \to \tau_{\leq 2}\mathbf{\Pi}(A) \to \tau_{\leq 1}\mathbf{\Pi}(A) \to \tau_{\leq 0}\mathbf{\Pi}(A) = *$

the intrinsic Postnikov tower of its path ∞-groupoid, the pasting composite of (∞,1)-pullbacks

$\array{ \vdots && && && \vdots \\ \downarrow && && && \downarrow \\ A_2 && &\to& \cdots &\to& \mathbf{B}\mathbf{\pi}_3(A) &\to& * \\ \downarrow && && && && \downarrow \\ A_1 && &\to& \cdots && && \mathbf{B}\mathbf{\pi}_2(A) &\to& * \\ \downarrow && && && && \downarrow && \downarrow \\ A &\to& \mathbf{\Pi}A &\to& \cdots &\to& \tau_{\leq 3} \mathbf{\Pi}A &\to& \tau_{\leq 2} \mathbf{\Pi}A &\to& \tau_{\leq 1} \mathbf{\Pi}A } \,,$

$* \to \cdots \to A_3 \to A_2 \to A_1 \to A_0 = A$

of $A$.

Since our $\mathbf{H}$ is assumed to be even ∞-connected, the Postnikov tower of $\mathbf{\Pi}(A)$ is the image under $LConst : \infty Grpd \to \mathbf{H}$ of the ordinary Postnikov tower of $\Pi(A)$ in $\infty Grpd$. Accordingly, we have $\mathbf{B} \mathbf{\pi}_n(A) = LConst B^n \pi_n \Pi(A)$

The point now is that in $\mathbf{H} =$ ∞LieGrpd we may form smooh refinements of these discrete extensions: every discrete $(n+1)$-group $\mathbf{B}^{n+1}\mathbb{Z}$ we want to refine to a smooth $n$-group $\mathbf{B}^n U(1)$. By the discussion at geometric realization, both have equivalent underlying $\infty$-groupoids

$\Pi(\mathbf{B}^{n+1}\mathbb{Z}) \simeq \Pi(\mathbf{B}^n U(1)) \simeq K(\mathbb{Z},n+1) \,.$

For every direct summand abelian group $\mathbb{Z}$ in one of the $\mathbf{\pi}_n(A)$ we can ask for a refinement of the cocycle from coefficients $\mathbf{B}^n \mathbb{Z}$ to $\mathbf{B}^{n-1}\mathbb{R}/\mathbb{Z}$. 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 $curv : \mathbf{B}^{n-1}\mathbb{R}/\mathbb{Z} \to \mathbf{\flat}_{dR}\mathbf{B}^{n}\mathbb{R}$.

For instance for $A = \mathbf{B}Spin \in \mathbf{H} = \infty Lie Grpd$ the delooping of the spin group, we refine the internal Whitehead tower to

$\array{ \vdots \\ \mathbf{B}Fivebrane &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}String &\to& \mathbf{B}^7 U(1) &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}Spin &\to& \cdots &\to& \mathbf{B}^3 U(1) } \,,$

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

$\mathbf{B}Spin \to \mathbf{B}^3 U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^4 U(1) = \mathbf{\flat}_{dR} \mathbf{B}^4 \mathbb{R} \,.$

If we have a $Spin$-principal bundle $X \to \mathbf{B}G$, we may form over it the covering circle $n$-group bundles on which these higher cocycles naturally live

$\array{ P_2 &\to& \mathbf{B}Fivebrane &\to& * \\ \downarrow && \downarrow && \downarrow \\ P_1 &\to& \mathbf{B}String &\to& \mathbf{B}^7 U(1) &\to& * \\ \downarrow && \downarrow && && \downarrow \\ X &\to& \mathbf{B}Spin &\to& \cdots &\to& \mathbf{B}^3 U(1) } \,.$

Here for $X$ an ordinary space, $X = \tau_0 X$, the higher circle $n$-group principal bundes $P_k$ have the property that also $\tau_0 P_k = X$. Therefore the 0-truncation of the entire composite

$P_1 \to \mathbf{B}^7 U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^8 \mathbb{R}$

defines a closed 8-form on $X$. 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 $X$, but on the covering $P_1$ of $X$.

(…)

## Examples

### 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 $G$-principal bundles and how the ∞-Chern-Weil homomorphism reduces on these to the ordinary Chern-Weil homomorphism.

Let $\mathfrak{g}$ be a (finite dimensional) Lie algebra. Then

$\mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G$

is the delooping of the simply connected Lie group $G$ integrating $\mathfrak{g}$.

###### Proposition

The coefficient object $\mathbf{B}G_{conn}$ of genuine ∞-Lie algebra connections for $\mathfrak{g}$ an ordinary Lie algebra is weakly equivalent to the simplicial presheaf

$\mathbf{B}G_{conn} \stackrel{\simeq}{\to} \Xi[G\times \Omega^1(-,\mathfrak{g}) \stackrel{ \overset{Ad_{p_1}(p_2)+ p_1 d p_1^{-1}}{\to}}{\underset{p_2}{\to}} \Omega^1(-,\mathfrak{g})]$

that assigns objectwise the groupoid of Lie-algebra valued 1-forms.

This is moreover isomorphic to the simplicial presheaf

$\cdots = [CartSp^{op},sSet](\mathbf{P}_1(-),\mathbf{B}G])$

of morphisms out of the path groupoid.

The flat coefficient object $\mathbf{\flat}\mathbf{B}G$ is modeled by the subobject

$\Xi[G\times \Omega^1_{flat}(-,\mathfrak{g}) \stackrel{ \overset{Ad_{p_1}(p_2)+ p_1 d p_1^{-1}}{\to}}{\underset{p_2}{\to}} \Omega^1_{flat}(-,\mathfrak{g})]$

of groupoids of Lie-algebra valued forms with vanishing curvature 2-form.

This is isomorphic to

$\cdots = [CartSp^{op},sSet](\mathbf{\Pi}_1(-),\mathbf{B}G])$

of morphism out of the fundamental groupoid.

###### Proof

The statements about morphisms out of the path groupoid are discussed in detail in SchrWalI.

###### Corollary

For $X$ a paracompact smooth manifold and $\{U_i \to X\}$ a good open cover we have a natural equivalence of groupoids

$[CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}G_{conn}) \simeq G Bund_\nabla(X)$

with the groupoid of smooth $G$-principal bundles with connection on $G$.

For $\langle - \rangle : inn(\mathfrak{g}) \to b^p \mathbb{R}$ an invariant polynomial on $\mathfrak{g}$, the induced morphism

$[CartSp^{op}, sSet](C(\U_i\}), \mathbf{B}G_{diff}) \stackrel{\langle F_{(-)}\rangle}{\to} \Omega^{p+1}_{cl}(X)$

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

$\langle F_{(-)}\rangle : \mathbf{H}(-, \mathbf{B}G) \to \mathbf{H}(-, \tau_{1} \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R} )$

in $\mathbf{H} = \infty LieGrpd$ represented by

$\array{ [CartSp^{op},sSet](-, \mathbf{B}G_{diff}) &\to& [CartSp^{op},sSet](-, \mathbf{cosk}_2 \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R}) \\ \downarrow^{\simeq} \\ [CartSp^{op},sSet](-, \mathbf{B}G) } \,.$

For $X$ a smooth manifold with good cover $\{U_i \to X\}$ we have that $[CartSp^{op},sSet](C\{U_i\}, \mathbf{cosk}_2 \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R})$ is the groupoid whose objects are closed $p+1$-forms on $X$ and whose morphisms are given by $p$-forms modulo exact forms.

Let $i \in I$ range over a set of generators for all invariant polynomials. Then

$\mathbf{H}(-,\mathbf{B}G) \to \prod_i \tau_1\mathbf{\flat}_{dR}\mathbf{B}^{n_i} \mathbb{R}$

is an approximation to the intrinsic Chern-character. We may consider its homotopy fibers over a given set $Q_i$ of curvature characteristic forms.

Assume $\nabla, \nabla' : C(\{U_i\}) \to \mathbf{B}G_{diff}$ are two genuine connections with coinciding curvature characteristic classes $\{Q_i\}$. Then in the homotopy fiber they are coboundant cocycles precisely if all the Chern-Simons forms $CS_i(\nabla,\nabla')$ 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 $\mathfrak{g}$ be a Lie strict 2-group coming from a differential crossed module $(\mathfrak{g}_2 \to \mathfrak{g}_1)$. Then we have two candidate Lie 2-groups integrating this: on the one hand the strict 2-group coming from the crossed module $(G_2 \to G_1)$ that integrates $(\mathfrak{g}_2 \to\mathfrak{g}_1)$ degreewise as ordinary Lie algebras, and on the other hand $cosk_k\exp(\mathfrak{g})$.

###### Proposition

The morphism

$\tau_2 \exp(\mathfrak{g}) \to \mathbf{B}(G_2 \to G_1)$

given by evaluating 2-dimensional parallel transport is a weak equivalence.

###### Proof

Use the 3-dimensional nonabelian Stokes theorem from the appendix of SchrWalII.

###### Corollary

The object $\mathbf{B}(G_2 \to G_1)_{conn}$ assigns to $U \in CartSp$ the 2-groupoid of Lie 2-algebra valued forms over $U$.

This is described in detail in SchrWalII, subject to the extra constraint that the 2-form curvature vanishes.

###### Corollary

A genuine connection on a $(G_2 \to G_1)$-principal 2-bundle with given cocycle $X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}(G_2 \to G_1)$ is a cocycle $X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}(G_2 \to G_1)_{conn}$ given as follows:

1. on $U_i$ a pair of forms $A_i \in \Omega^1(U_i, \mathfrak{g}_1)$, $B_i \in \Omega^2(U_i, \mathfrak{g}_2)$;

1 on $U_i \cap U_j$ a function $g_{i j} \in C^\infty(U_{i}\cap U_j , G_1)$ and a 1-form $a_{i j} \in \Omega^1(U_i \cap U_j, \mathfrak{g}_2)$ such that

1. and so forth

This is described in detail in SchrWalIII, subject to the extra constraint that the 2-form curvature vanishes.

(…)

### Twisted differential $String-$ and $Fivebrane$-structures

We discuss now in detail refined Chern-Weil morphisms

$\hat \mathbf{c} : \mathbf{H}_{conn}(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X, \mathbf{B}^n U(1))$

that send ∞-connections on $G$-principal ∞-bundles to circle n-bundles with connection that represent a given characteristic class. $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ 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 $O$ that are controled by ∞-Lie algebra cohomology.

The smooth Whitehead tower of $O$ in ∞LieGrpd starts as

$\cdots \to \mathbf{B}Spin \to \mathbf{B} SO \to \mathbf{B}O \,,$

where

• the delooping $\mathbf{B} SO$ of the special orthogonal group is the $\mathbb{Z}_2$-principal bundle over $\mathbf{B}O$ classified by the cocycle $\mathbf{B}O \to \mathbf{B} \mathbb{Z}_2$ that sends an elemen $k \in O$ to $+1$ if it is in the connected component of the identity and to $-1$ if it is not. This means we have an (∞,1)-pullback diagram

$\array{ \mathbf{B} SO &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} O &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B} \mathbb{Z}_2 } \;$
• the delooping $\mathbf{B} Spin$ of the spin group is the is the $\mathbf{B}\mathbb{Z}_2$-principal 2-bundle over $\mathbf{B} SO$ classified by the Stiefel-Whitney class $\mathbf{B} SO \to \mathbf{B}^2 \mathbb{Z}$

$\array{ \mathbf{B} Spin &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \;$

Since these two steps are controled by the torsion-group $\mathbb{Z}_2$ they have no nontrivial refinement to differential cohomology. The next step however is controled by what in the (∞,1)-topos ∞Grpd $\simeq$ Top is the first fractional Pontryagin class? $\frac{1}{2}p_1 : \mathcal{B} Spin \to \mathcal{B}^4 \mathbb{Z}$ and which lifts through the path ∞-groupoid functor $\Pi : \infty LieGrpd \to \infty Grpd$ to a characteristic class in $\mathbf{H} =$ ∞LieGrpd (as discussed there) $\frac{1}{2} p_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)$ with coefficients in the smooth circle 3-groupoid. This cocycle does arise as the Lie integration $\exp(\mu)$ of the canonical Lie algebra 3-cocycle $\mu = \langle -,[-,-]\rangle: \mathfrak{so} \to b^2 \mathbb{R}$.

The principal 3-bundle that this classifies is the delooping $\mathbf{B} String$ of the string 2-group $String$

$\array{ \mathbf{B} String &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) } \,.$

Notice that the fact that this is an (∞,1)-pullback implies that for any $X\in \mathbf{H} = \infty LieGrpd$ also

$\array{ \mathbf{H}(X,\mathbf{B} String) &\to& * \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B} Spin) &\to& \mathbf{H}(X,\mathbf{B}^3 U(1)) } \,,$

which exhibits the 2-groupoid $\mathbf{H}(X,\mathbf{B}String)$ 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 $3$-bundle with connection

We describe the special case of the general $\infty$-Chern–Weil homomorphism for $\infty$-Lie algebra valued connections corresponding to the characteristic class $\frac{1}{2}p_1\colon \mathbf{B}Spin \to \mathbf{B}^3 U(1)$: the first fractional Pontryagin class of the spin group $\mathbf{B}Spin$. The $\mathbf{B}^3 U(1)$-differential cocycle that it produces from a given $Spin$-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 $\infty$-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

Let

$\frac{1}{6}\hat p_2 : \mathbf{H}_{conn}(-,\mathbf{B}String) \to \mathbf{H}_{diff}(-, \mathbf{B}^7 U(1))$

be the differential refinement of the second fractional Pontryagin class discussed above.

Definition

For $X \in \mathbf{H} =$ ∞LieGrpd, the $\infty$-groupoid of differential fivebrane-structures $Fivebrane_{diff}(X)$ is the homotopy fiber of $\frac{1}{6}p_2(X) : \mathbf{H}(X,\mathbf{B}String) \to \mathbf{H}_{diff}(X, \mathbf{B}^7 U(1))$.

More generally, the $\infty$-groupoid of twisted differential fivebrane structures is the (∞,1)-pullback $Fivebrane_{diff,tw}(X)$ in

$\array{ Fivebrane_{diff,tw}(X) &\to& H_{diff}(X,\mathbf{B}^7 U(1)) \\ \downarrow && \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}String) &\stackrel{\frac{1}{6}\hat p_2}{\to}& \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1)) } \,.$

In terms of the underlying $\infty$-Lie algebra valued local connection data, i.e. before Lie integration in the above sense , this has been considered in SSSIII

(…)

### $\infty$-Chern-Simons theory

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 $n$-dimensional curves $\Sigma_n \to X$. 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

## References

An explicit presentation of the $\infty$-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

Revised on August 20, 2011 20:59:16 by Urs Schreiber (89.204.153.107)