In an associated bundle with connection the covariant derivative of a section is a measure for how that section fails to be constant with respect to the connection.
We give here a definition of covariant derivatives that is natural in the general context of ∞-Chern-Weil theory in that it applies to connections on ∞-bundles.
We start by describing this just for ordinary connections on a bundle and demonstrate how this general abstract definition reproduces the traditional definitions found in the literature.
The central statement is: a covariant derivative $\nabla \sigma$ of a section may be identified with the 1-form curvature-component of a Lie algebroid-valued connection, and the curvature equation
is the Bianchi identity on its curvature 1-form.
Let $G$ be a Lie group, $V$ a smooth manifold and $\rho : G \times V \to V$ a smooth action. Write $V//G$ for the corresponding action groupoid, itself a Lie groupoid. The Lie algebroid $Lie(V//G)$ corresponding to this is the action Lie algebroid.
Below we shall define covariant derivatives as curvature components of ∞-Lie algebroid valued forms with values in this action Lie algebroid. To prepare the ground for this, the following observation recalls some basic facts.
The Chevalley-Eilenberg algebra of the action Lie algebroid is
where the differential acts on functions $f \in C^\infty(V)$ by
Explicitly, for $t \in \mathfrak{g}$ this sends $f$ to the function $(d_\rho f)(t)$ which is the derivative along $t \in T_e G$ of the function $G \times V \stackrel{\rho}{\to}V \stackrel{f}{\to} \mathbb{R}$.
Even more explicitly, if we choose local coordinates $\{v^k\} : \mathbb{R}^{dim V} \to V$ on a patch, and choose a basis $\{t^a\}$ of $\mathfrak{g}^*$ then we have that restricted to this patch the differential is on generators given by
Specifically for $V$ a finite dimensional vector space, $\rho : G$ a linear action, $\{v^k\}$ a choice of basis of that vector space and $f$ a linear function $f= f_k v^k$ , we have that $(f_k := \partial_k f) \in \mathbb{R}^{dim V}$ are the components vector of the dual vector given by $V$ in this basis, and the above gives the matrix multiplication form of the action
Notice for completeness that the equation $(d_\rho)^2 = 0$ is equivalent to the Jacobi identity of the Lie bracket and the action property of $\rho$:
These local formulas shall be useful below for recognizing from our general abstract definition of covariant derivative the formulas traditionally given in the literature. For that notice that in the above local coordinates further restricting attention to linear actions, the Weil algebra of the action Lie algebroid is given by
where the differential is given on generators by
and where the uniquely induced differential on the shifted generators – the one encoding Bianchi identities – is
and
Notice that we may identify the delooping Lie groupoid $\mathbf{B}G$ of $G$ with the action groupoid of the trivial action on the point, $\mathbf{B}G \simeq *//G$. On Lie algebroids this morphism is dually the inclusion
that is the identity on $\mathfrak{g}^*$.
For $X$ smooth manifold, a $G$-principal bundle $P \to X$ is given by a cocycle in ∞LieGrpd $g : X \to \mathbf{B}G$.
The sections $\sigma$ of the corresponding $\rho$-associated bundle $P \times_\rho V$ are in natural bijection with the lifts to a $V//G$-cocycle
We may model the coycle $X \to \mathbf{B}G$ in the model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj,loc}$ by an anafunctor $X \stackrel{\simeq}{\leftarrow} C(U) \stackrel{g}{\to} \mathbf{B}G$ for $C(U)$ the Cech groupoid of a good open cover $\{U_i \to X\}$. This is a collection of smooth functions $(g_{i j} : U_i \cap U_j \to G)$ such that
A lift of this cocycle through $V//G \to \mathbf{B}G$ is in addition a collection of smooth functions $\{sigma_i : U_i \to V\}$ such that on all $U_i \cap U_j$ the equation
is satisfied. This identifies the $\sigma_i$ as precisely the components of a section $\sigma$ of $P \times_\rho G$ with respect to the local trivialization encoded by $g$.
Given a connection $\nabla$ on the $G$-principal bundle with cocycle $g$, there is a unique connection $\nabla \sigma$ on the $V//G$-groupoid principal bundle that corresponds to a section $\sigma$ by the above proposition.
The covariant derivative of a section $\sigma$ is the 1-form component $F_{\nabla \sigma}^1$ of the curvature of this groupoid-bundle connection.
This 1-form curvature is literally the measure for the non-flatness of the section . Whereas the 2-form curvature is a measure for the non-flatness of the connection.
We unwind this definition and find the traditional formulation of covariant derivatives as traditionally stated in the literature.
On a patch $U_i \hookrightarrow X$ the connection $\nabla$ is given by a morphism of dg-algebras
for $W(\mathfrak{g})$ the Weil algebra of $\mathfrak{g}$.
The groupoid connection $\nabla \sigma$ on this patch is given by
In degree 0 this is an algebra homomorphism
This is the dual of the local section $\sigma_i$ itself. In the case that $V$ is a vector space with chosen basis $\{v^k\}$, we have the corresponding components $(\sigma_i^k)$ of the local section.
Further, in degree 1 the connection is linear map
which is the connection form itself, as well as a linear map
which is the curvature 1-form. The respect of these maps for the differential says that
This is the familiar local formula for a covariant derivative as one finds it in the literature. We therefore write for short
If we keep $\nabla$ fixed and let $\sigma_i$ vary, then this may be thought of as a 1-form with values in endomorphisms of the space of sections
There is a Bianchi identity on every curvature component, induced from the respect for differentials of the dg-algebra morphism $\Omega^\bullet(U_i) \leftarrow W(Lie(V//G)) : F_\nabla$ on shifted generators.
From the discussion at Action Lie algebroid cohomology above we read off the Bianchi identity for the 1-form curvature that we identified with the covariant derivative in the case of linear actions to be given in local coordinates (as above) by (suppressing the patch index ${}_i$)
More invariantly we may write this as
and this find the usual expression of the curvature of a connection as the square of the covariant derivative.
gauge field: models and components
A discussion of covariant derivatives for a Levi-Civita connection in terms of synthetic differential geometry is in