# nLab Chern-Weil homomorphism

Contents

### Context

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

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

## Idea

For $G$ a Lie group with Lie algebra $\mathfrak{g}$, a $G$-principal bundle $P \to X$ on a smooth manifold $X$ induces a collection of classes in the de Rham cohomology of $X$: the classes of the curvature characteristic forms

$\langle F_A \wedge \cdots F_A \rangle \in \Omega^{2n}_{closed}(X)$

of the curvature 2-form $F_A \in \Omega^2(P, \mathfrak{g})$ of any connection on $P$, and for each invariant polynomial $\langle -\rangle$ of arity $n$ on $\mathfrak{g}$.

This is a map from the first nonabelian cohomology of $X$ with coefficients in $G$ to the de Rham cohomology of $X$

$char : H^1(X,G) \to \prod_{n_i} H_{dR}^{2 n_i}(X)$

where $i$ runs over a set of generators of the invariant polynomials. This is the analogy in nonabelian differential cohomology of the generalized Chern character map in generalized Eilenberg-Steenrod-differential cohomology.

## Plain Chern-Weil homomorphism

This subsection is to give an outline of construction of Weil homomorphism as in Kobayashi-Nomizu 63

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Given an element $g\in G$, the adjoint map $Ad(g):G\rightarrow G$ is defined as $Ad(g)(h)=ghg^{-1}$. For $g\in G$, let $ad(g):\mathfrak{g}\rightarrow \mathfrak{g}$ be the differenial of $Ad(g):G\rightarrow G$ at $e\in G$.

Let $I^k(G)$ denote the set of symmetric, multilinear maps

$f:\underbrace{\mathfrak{g}\times\cdots\times\mathfrak{g}}_{k ~\text{times}}\rightarrow \mathbb{R}$

that are $G$ invariant in the sense that $f(ad(g)(t_1),\cdots,ad(g)(t_k))=f(t_1,\cdots,t_k)$ for all $g\in G$ and $t_i\in \mathfrak{g}$. These $I^k(G)$ are vector spaces over $\mathbb{R}$. Let $I(G)$ denote the $\mathbb{R}$ algebra $\oplus_{k=0}^{\infty}I^k(G)$.

Let $M$ be a manifold and $H^*(M,\mathbb{R})$ be the deRham cohomology ring of $M$.

Given a principal $G$ bundle over $M$, say $\pi:P\rightarrow M$, Weil homomorphism gives a homomorphism $I(G)\rightarrow H^*(M,\mathbb{R})$. Though it does not depend on connection on $P(M,G)$, the construction of this map is done after fixing a connection on $P(M,G)$. Outline of the construction is as follows.

1. Fix a connection $\Gamma$ on $P(M,G)$. Let $\Omega$ denote the curvature of $\Gamma$.

2. Given an element $f\in I^k(G)$, define a $2k$-form $f(\Omega)$on $P$. \item Prove that the $2k$ form $f(\Omega)$ on $P$ projects uniquely to a $2k$ form on $M$ and call it $\tilde{f}(\Omega)$ i.e., $\pi^*(\tilde{f}(\Omega))=f(\Omega)$.

3. Next step is to prove that $\tilde{f}(\Omega)$ is closed $2k$ form on $M$. To prove $\tilde{f}(\Omega)$ is closed, it suffices to prove that $f(\Omega)$ is closed.

4. For a special $k$-form $\varphi$ on $P$, the exterior differential $d\varphi$ coincides with the exterior covariant differential $D\varphi$ of $\varphi$ i.e., $d\varphi=D\varphi$. That special property is that $\varphi=\pi^*\sigma$ for some $k$-form $\sigma$ on $M$.

5. As $f(\Omega)$ has that special property, we see that $d(f(\Omega))=D(f(\Omega))$.

6. By Bianchi’s identity, we have $D\Omega=0$. We then see that $D\Omega=0$ implies that $D(f(\Omega))=0$ i.e., $d(f(\Omega))=D(f(\Omega))=0$ for $f\in I^k(G)$ i.e., $f(\Omega)$ is a closed $2k$-form on $P$. Thus, $\tilde{f}(\Omega)$ is a closed $2k$-form on $M$, giving an element in the deRham cohomology $H^{2k}(M,\mathbb{R})$.

7. Next step is to prove that, this assignment $f\mapsto \tilde{f}(\Omega)$ does not depend on the connection $\Gamma$ that we have started with i.e., for connections $\Gamma_0$ (with curvature form $\Omega_0$) and $\Gamma_1$ (with curvature form $\Omega_1$), the elements $\tilde{f}(\Omega_0)$ and $\tilde{f}(\Omega_1)$ are in the same equivalence class i.e., $\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)$ is an exact form i.e., $\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)=d\tilde{\Phi}$ for some $2k-1$ form $\tilde{\Phi}$ on $M$.

8. Using lemma , to prove $\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)=d\tilde{\Phi}$ for some $2k-1$ form $\tilde{\Phi}$ on $M$, it suffices to prove that $f(\Omega_0)-f(\Omega_1)=d \Phi$ for some $2k-1$ form $\Phi$ on $P$ that projects to a unique $2k-1$ form $\tilde{\Phi}$ on $M$.

9. We then see that $f(\Omega_0)-f(\Omega_1)=d \Phi$ for some $2k-1$ form $\Phi$ on $P$ that projects to a unique $2k-1$ form $\tilde{\Phi}$ on $M$. This confirm that the assignment $f\mapsto f(\Omega)$ is independent of the connection $\Gamma$ that we have started with. We can extend this linearly to $I(G)\rightarrow H^*(M,\mathbb{R})$.

Given a principal bundle $\pi:P\rightarrow M$ the morphism defined above $I(G)\rightarrow H^*(M,\mathbb{R})$ is called the Weil homomorphism.

## Refined Chern-Weil homomorphism

We describe the refined Chern-Weil homomorphism (which associates a class in ordinary differential cohomology to a principal bundle with connection) in terms of the universal connection on the universal principal bundle. We follow (HopkinsSinger, section 3.3).

• Let $G$ be a compact Lie group

• with Lie algebra $\mathfrak{g}$;

• and write $inv(\mathfrak{g})$ for the dg-algebra of invariant polynomials on $\mathfrak{g}$ (which has trivial differential).

• Write $B^{(n)}G$ for the smooth level $n$ classifying space

• and $B G := {\lim_\to}_n B^{(n)}G$ for the colimit, a smooth model of the classifying space of $G$.

• Write $\nabla_{univ}$ for the universal connection on $E G \to B G$.

• Let $[c] \in H^k(B G, \mathbb{Z})$ be a characteristic class

• and choose a refinement $[\hat \mathbf{c}] \in H_{diff}^k(B G)$ in ordinary differential cohomology represented by a differential function

$(c, h, w) \in C^k(B G, \mathbb{Z}) \times C^{k-1}(B G, \mathbb{R}) \times (W(\mathfrak{g}) \simeq C^k(B G, \mathbb{R}))^k \,.$
###### Definition

For $X$ a smooth manifold, $P \to X$ a smoth $G$-principal bundle with smooth classifying map $f : X \to B G$ and connection $\nabla$. Write $CS(\nabla, f^* \nabla_{univ})$ for the Chern-Simons form for the interpolation between $\nabla$ and the pullback of the universal connection along $f$.

Then defined the cocycle in ordinary differential cohomology given by the function complex

$\hat \mathbf{c} := (f^* c , f^* h + CS(\nabla, f^* \nabla_{univ}), w(F_{\nabla_t})) \in (c, h, w) \in C^k(B G, \mathbb{Z}) \times C^{k-1}(B G, \mathbb{R}) \times \Omega_{cl}^k(X) \,.$
###### Proposition

The above construction constitutes a map

$\hat \mathbf{c} : G Bund_\nabla(X)_\sim \to H_{diff}^k(X)$

from equivalence classes of $G$-principal bundles with connection to degree $k$ ordinary differential cohomology.

## References

### Chern-Weil homomorphism

#### Original articles

The differential-geometric Chern-Weil homomorphism (evaluating curvature 2-forms of connections in invariant polynomials) first appears in print (_Cartan's map) in

• Henri Cartan, Section 7 of: Cohomologie réelle d’un espace fibré principal différentiable. I : notions d’algèbre différentielle, algèbre de Weil d’un groupe de Lie, Séminaire Henri Cartan, Volume 2 (1949-1950), Talk no. 19, May 1950 (numdam:SHC_1949-1950__2__A18_0)

Henri Cartan, Section 7 of: Notions d’algèbre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie, in: Centre Belge de Recherches Mathématiques, Colloque de Topologie (Espaces Fibrés) Tenu à Bruxelles du 5 au 8 juin 1950, Georges Thon 1951 (GoogleBooks, pdf)

reprinted in the appendix of:

(These two articles have the same content, with the same section outline, but not the same wording. The first one is a tad more detailed. The second one briefly attributes the construction to Weil, without reference.)

and around equation (10) of:

• Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf, full proceedings vol 2 pdf)

It is the independence of this construction under the choice of connection which Chern 50 attributes (below equation 10) to the unpublished

• André Weil, Géométrie différentielle des espaces fibres, unpublished, item [1949e] in: André Weil Oeuvres Scientifiques / Collected Papers, vol. 1 (1926-1951), 422-436, Springer 2009 (ISBN:978-3-662-45256-1)

The proof is later recorded, in print, in: Chern 51, III.4, Kobayashi-Nomizu 63, XII, Thm 1.1.

But the main result of Chern 50 (later called the fundamental theorem in Chern 51, XII.6) is that this differential-geometric “Chern-Weil” construction is equivalent to the topological (homotopy theoretic) construction of pulling back the universal characteristic classes from the classifying space $B G$ along the classifying map of the given principal bundle.

This fundamental theorem is equation (15) in Chern 50 (equation 31 in Chern 51), using (quoting from the same page):

methods initiated by E. Cartan and recently developed with success by H. Cartan, Chevalley, Koszul, Leray, and Weil [13]

Here reference 13 is:

More in detail, Chern’s proof of the fundamental theorem (Chern 50, (15), Chern 51, III (31)) uses:

1. the fact that invariant polynomials constitute the real cohomology of the classifying space, $inv(\mathfrak{g}) \simeq H^\bullet(B G)$, which is later expanded on in:

Some authors later call this the “abstract Chern-Weil isomorphism”.

2. existence of universal connections for manifolds in bounded dimension (see here), which is later developed in:

#### Review

Review of the Chern-Weil homomorphism:

With an eye towards applications in mathematical physics: