# nLab Ehresmann connection

### Context

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

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

∞-Lie theory

# Contents

## Idea

The notion of Ehresmann connection is one of the various equivalent definitions of connection on a bundle.

## Definition

### In terms of differential forms

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and $P \to X$ a $G$-principal bundle. Let

$\rho : P \times G \to P$

be the action of $G$ on $P$ and

$\rho_* : \mathfrak{g} \to \Gamma(T X)$

its derivative, sending each element $x \in \mathfrak{g}$ to the vector field on $P$ that at $p \in P$ is the push-forward $\rho(p,-)_*(x)$.

For $v \in \Gamma(T X)$ and $\omega$ a differential form on $P$ write $\iota_v \omega$ for the contraction.

###### Definition

A Cartan-Ehresmann connection on $P$ is a Lie algebra-valued 1-form

$A\in \Omega^1(P, \mathfrak{g})$

on $P$ satisfying two conditions

1. first Ehresmann condition

for every $x \in \mathfrak{g}$ we have

$\iota_{\rho_*(x)} A = x \,.$
2. second Ehresmann condition

for every $x \in \mathfrak{g}$ we have

$\mathcal{L}_{\rho_*(x)} A = ad_x A \,,$

where $\mathcal{L}_{\rho_*(x)}$ is the Lie derivative along $\rho_*(x)$ and where $ad_x : \mathfrak{g} \to \mathfrak{g}$ is the adjoint action of $\mathfrak{g}$ on itself.

###### Proposition

This is equivalent to

1. first Ehresmann condition

for every $x \in \mathfrak{g}$ we have

$\iota_{\rho_*(x)} A = x \,.$
2. second Ehresmann condition

for every $x \in \mathfrak{g}$ we have

$\iota_{\rho_*(x)} F_A = 0 \,,$

where $F_A \in \Omega^2(P, \mathfrak{g})$ is the curvature 2-form of $A$.

###### Proof

Using $\iota_x A = x$ we have by Cartan calculus

\begin{aligned} \iota_{\rho_*(x)} F_A &= \iota_{\rho_*(x)} d_{dR} A + \frac{1}{2}[A\wedge A] \\ & = \mathcal{L}_{\rho_*(x)} A - d_{dR} \iota_{\rho_*(x)} A + [x,A] \\ & = \mathcal{L}_{\rho_*(x)} A + [x,A] \,. \end{aligned}

### In terms of distributions

Given a smooth bundle $\pi: E\to X$ with typical fiber $F$ (e.g. a smooth vector bundle or a smooth principal $G$-bundle), there is a well defined vector subbundle $V E\subset T E$ over $E$ such that $V_p$ consists of the tangent vectors $v_p$ such that $(T_p \pi)(v_p) = 0$. A smooth distribution (field) of horizontal subspaces is a choice of a vector subspace $H_p E\subset T_p E$ for every $p$ such that

E1. (complementarity) $T_u E \: = \: H_u E \oplus V_u E$

E2. $p\mapsto H_p E$ is smooth. That means that in the unique decomposition of any smooth vector field $X$ on $E$ into vector fields $X^H \in \Gamma(H_u E)$ and $X^V \in \Gamma(V_u E)$ such that $X = X^H + X^V$ the vector field $X^H$ is smooth (or equivalently $X^V$ is smooth, or equivalently both) as a section of $T E$ (there exist yet several other equivalent formulations of the smooothness criterion).

An Ehresmann connection describes a connection on a $G$-principal bundle $\pi : P \to X$ (for $G$ some Lie group) in terms of a distribution of horizontal subspaces $H \subset T P$ which is a subbundle of the tangent bundle of $P$ complementary at each point to the vertical tangent bundle to the fiber. More precisely, an Ehresmann connection on a principal $G$-bundle $\pi:P\to X$ is a smooth distribution of horizontal subspaces $p\mapsto H_p P$ which is equivariant:

E3. $H_{p g}P = (T_p R_g) H_p P$ for every $p \in P$ and $g \in G$.

This subbundle $H = \cup_p H_p\subset T X$ over $X$ can be expressed as a field of subspaces $H_x = Ker A_x = Ann A_x\subset T P$ ($x\in P$) which are pointwise annihilators of a smooth Lie algebra-valued $1$-form $A \in \Omega^1(P,Lie(G))$ on $P$ that satisfies two Ehresmann conditions from the previous subsection.

The Ehresmann connections on a principal $G$-bundle are in 1-1 correspondence with an appropriate notion of a connection on the associated bundle. Namely, if $T^H P\subset T P$ is the smooth horizontal distrubution of subspaces defining the principal connection on a principal $G$-bundle $P$ over $X$, where $G$ is a Lie group and $F$ a smooth left $G$-space, then consider the total space $P\times_G F$ of the associated bundle with typical fiber $F$. Then, for a fixed $f\in F$ one defines a map $\rho_f : P\to P\times_G F$ assigning the class $[p,f]$ to $p\in P$. If $(T_p \rho_f)(T^H_p P) =: T_{[p,f]}^H P\times_G F$ defines the horizontal subspace $T_{[p,f]}^H P\times_G F\subset T_{[p,f]} P\times_G F$, the collection of such subspaces does not depend on the choice of $(p,f)$ in the class $[p,f]$, and the correspondence $p\mapsto T_{[p,f]}^H P\times_G F$ is a connection on the associated bundle $P\times_G F\to X$.

### In terms of cohesive homotopy type theory

One may also describe(flat) Ehresmann connections in cohesive homotopy type theory.

The general abstract discussion is here. The discussion of how in smooth infinity-groupoids this reduces to the traditional notion is here.

## Properties

### General

###### Definition

The two definitions in terms of 1-forms and in terms of horizontal distributions are equivalent.

###### Proof

At each $p \in P$ take the horizontal subspace $H_p P$ to be the kernel of $A(p) : T_p P \to \mathfrak{g}$

$H_p P := ker A(p)$
###### Remark

This means we may think of $A$ as measuring how infinitesimal paths in $P$ fail to be horizontal or parallel to $X$ in the sense of parallel transport.

### Curvature characteristic forms

Let $\langle - \rangle \in W(\mathfrak{g})$ be an invariant polynomial on the Lie algebra. For $A \in Omega^1(P,\mathfrak{g})$ an Ehresmann connection, write

$\langle F_A \rangle = \langle F_A \wedge F_A \wedge \cdots F_A\rangle$

for the curvature characteristic form obtained by evaluating this on wedge powers of the curvature 2-form.

###### Proposition

The forms $\langle F_A \rangle \in \Omega^{2k}(P)$ are closed, descend along $p : P \to X$, in that they are pullbacks of forms along $p$, and their class in de Rham cohomology $H^{2k}(X)$ are independent of the choice of $A$ on $P$.

###### Proof

That the forms are closed follows from the Bianchi identity

$d F_A = [A\wedge F_A]$

satisfied by the curvature 2-form and the defining as-invariance of $\langle-\rangle$. More abstractly, the 1-form $A$ itself may be identified with a morphism of dg-algebras out of the Weil algebra $W(\mathfrak{g})$ (see there)

$\Omega^\bullet(P) \leftarrow W(\mathfrak{g}) : A$

and the evaluation of the curvature in the invariant polynomials corresponds to the precomposition with the morphism

$W(\mathfrak{g}) \leftarrow CE(b^{2k-1}\mathbb{R} ) : \langle - \rangle$

described at ∞-Lie algebra cohomology.

to show that these forms descend, it is sufficient to show that for all $x \in \mathfrak{g}$ we have

1. $\iota_{\rho_*(X)} \langle F_A \rangle = 0$

2. $\mathcal{L}_{\rho_*(x)} \langle F_A \rangle = 0$

The first follows from $\iota_{\rho_*(x)} F_A = 0$. The second from this, the $d_{dR}$-closure just discussed and Cartan's magic formula for the Lie derivative.

###### Remark

The form $\langle F_A \rangle$ is called the curvature characteristic form of the connection $A$. The map

$\inv(\mathfrak{g}) \to \Omega^\bullet(X)$

induced by $(P,A)$ as above is the Chern-Weil homomorphism.

## Note on terminology

The terminology for the various incarnations of the single notion of connection on a bundle varies throughout the literature. What we here call an Ehresmann connection is sometimes, but not always, called principal connection (as it is defined for principal bundles).

## References

The original definition is due to

• Charles Ehresmann, Les connexions infinitésimale dans une espace fibré différentiable, Colloque de Topologie, Bruxelles (1950) 29-55, MR0042768