nLab
covariant derivative

Idea

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.

Definition

In the context of connections on $\mathrm{\infty }$-groupoid principal bundles

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

$$\nabla \nabla \sigma =F_{\nabla }\sigma$$\nabla \nabla \sigma = F_\nabla \sigma

is the Bianchi identity on its curvature 1-form.

Preliminaries on action Lie algebroid cohomology

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 $\mathrm{Lie}(V//G)$ 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

$$\mathrm{CE}(\mathrm{Lie}(V//G))=({\wedge }_{C^{\mathrm{\infty }}(V)}^{•}{𝔤}^{*},d_{\rho }),$$CE(Lie(V//G)) = (\wedge^\bullet_{C^\infty(V)} \mathfrak{g}^*, d_{\rho}) \,,

where the differential acts on functions $f\in C^{\mathrm{\infty }}(V)$ by

$$d_{\rho }:f\mapsto \rho (-)(-{)}^{*}f\in C^{\mathrm{\infty }}(V)\otimes {𝔤}^{*}.$$d_\rho : f \mapsto \rho(-)(-)^* f \in C^\infty(V)\otimes \mathfrak{g}^* \,.

Explicitly, for $t\in 𝔤$ this sends $f$ to the function $(d_{\rho }f)(t)$ which is the derivative along $t\in T_{e}G$ of the function $G\times V\stackrel{\rho }{\to }V\stackrel{f}{\to }ℝ$.

Even more explicitly, if we choose local coordinates $\{v^k\}:{ℝ}^{\dim V}\to V$ on a patch, and choose a basis $\{t^a\}$ of ${𝔤}^{*}$ then we have that restricted to this patch the differential is on generators given by

$$d_{\rho }:f\mapsto {\rho }^{\mu }{}_a{t}^a\wedge {\partial }_{k}f$$d_\rho : f \mapsto \rho^\mu{}_a t^a \wedge \partial_k f

$$d_{\rho }:t^a\mapsto -\frac{1}{2}{C}^a{}_{bc}{t}^b\wedge t^c.$$d_\rho : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,.

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 $(f_k:={\partial }_{k}f)\in {ℝ}^{\dim V}$ 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

$$d_{\rho }:v^k\mapsto t^a{\rho }_a{}^k{}_l{v}^l.$$d_\rho : v^k \mapsto t^a \rho_a{}^k{}_l v^l \,.

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 $\rho$:

$$d_{\rho }{d}_{\rho }{v}^k=(t^a\wedge t^b{\rho }_a{}^k{}_r{\rho }_b{}^r{}_l-\frac{1}{2}{C}^a{}_{bc}{t}^b\wedge t^c{\rho }_a{}^k{}_l)v^l.$$d_\rho d_\rho v^k = (t^a \wedge t^b \rho_a{}^k{}_r \rho_b{}^r{}_l - \frac{1}{2}C^a{}_{b c}t^b \wedge t^c \rho_a{}^k{}_l ) v^l \,.

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

$$W(\mathrm{Lie}(V//G))=({\wedge }_{C^{\mathrm{\infty }}({ℝ}^{\dim V})}^{•}(\Gamma (T^{*}{ℝ}^{\dim V})\oplus {𝔤}^{*}\oplus {𝔤}^{*}[1]),d_{W_{\rho }})$$W(Lie(V//G)) = (\wedge^\bullet_{C^\infty(\mathbb{R}^{dim V})} ( \Gamma(T^* \mathbb{R}^{dim V}) \oplus \mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W_\rho})

where the differential is given on generators by

$$d_{W_{\rho }}:v^k\mapsto {\rho }_a{}^k{}_l{t}^a\wedge v^l+d_{\mathrm{dR}}{v}^k$$d_{W_\rho} : v^k \mapsto \rho_a{}^k{}_l t^a \wedge v^l + d_{dR} v^k

$$d_{W_{\rho }}:t^a\mapsto -\frac{1}{2}{C}^a{}_{bc}{t}^b\wedge t^c+r^a$$d_{W_\rho} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a

and where the uniquely induced differential on the shifted generators – the one encoding Bianchi identities – is

$$d_{W_{\rho }}:d_{\mathrm{dR}}{v}^k\mapsto {\rho }_a{}^k{}_k{r}^a\wedge v^l-{\rho }_a{}^k{}_l{t}^a\wedge d_{\mathrm{dR}}{v}^l$$d_{W_\rho} : d_{dR} v^k \mapsto \rho_a{}^k{}_k r^a \wedge v^l - \rho_a{}^k{}_l t^a \wedge d_{dR} v^l

and

$$d_W:r^a\mapsto C^a{}_{bc}{t}^b\wedge r^c.$$d_{W} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c \,.

Notice that we may identify the delooping Lie groupoid $BG$ of $G$ with the action groupoid of the trivial action on the point, $BG\simeq *//G$. On Lie algebroids this morphism is dually the inclusion

$$\mathrm{CE}(\mathrm{Lie}(V//G))\leftarrow \mathrm{CE}(𝔤)$$CE(Lie(V//G)) \leftarrow CE(\mathfrak{g})

that is the identity on ${𝔤}^{*}$.

Sections of bundles as groupoid principal bundles

For $X$ smooth manifold, a $G$-principal bundle $P\to X$ is given by a cocycle in ∞LieGrpd $g:X\to BG$.

Proposition

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

$$\begin{array}{ccc}& & V//G\\ & {}^{\sigma }\nearrow & \downarrow \\ X& \stackrel{g}{\to }& BG\end{array}.$$\array{ && V//G \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G } \,.

Proof

We may model the coycle $X\to BG$ in the model structure on simplicial presheaves $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$ by an anafunctor $X\stackrel{\simeq }{\leftarrow }C(U)\stackrel{g}{\to }BG$ for $C(U)$ the Cech groupoid of a good open cover $\{U_i\to X\}$. This is a collection of smooth functions $(g_{ij}:U_i\cap U_j\to G)$ such that

$$\left(\begin{array}{ccc}& & (x,j)\\ & \nearrow & & \searrow \\ (x,i)& & & & (x,k)\end{array}\right)\mapsto \left(\begin{array}{ccc}& & •\\ & {}^{g_{ij}(x)}\nearrow & & {\searrow }^{g_{jk}(x)}\\ •& & \stackrel{g_{ik}(x)}{\to }& & •\end{array}\right).$$\left( \array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) &&&& (x,k) } \right) \mapsto \left( \array{ && \bullet \\ & {}^{\mathllap{g_{i j}(x)}}\nearrow && \searrow^{\mathrlap{g_{j k}(x)}} \\ \bullet &&\stackrel{g_{i k}(x)}{\to}&& \bullet } \right) \,.

A lift of this cocycle through $V//G\to BG$ is in addition a collection of smooth functions $\{{\mathrm{sigma}}_i:U_i\to V\}$ such that on all $U_i\cap U_j$ the equation

$${\sigma }_j=\rho (g_{ij})({\sigma }_i)$$\sigma_j = \rho(g_{i j})(\sigma_i)

$$\left(\begin{array}{ccccc}{\sigma }_i(x)& & \stackrel{\rho (g_{ik}(x))}{\to }& & {\sigma }_k(x)\end{array}\right)$$\left( \array{ \sigma_i(x) &&\stackrel{\rho(g_{i k}(x))}{\to}&& \sigma_k(x) } \right)

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 $g$.

Definition of covariant derivative

Observation

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.

Definition

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.

Equivalence to more traditional definitions

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

$${\Omega }^{•}(U_i)\leftarrow W(𝔤):A_i$$\Omega^\bullet(U_i) \leftarrow W(\mathfrak{g}) : A_i

for $W(𝔤)$ the Weil algebra of $𝔤$.

The groupoid connection $\nabla \sigma$ on this patch is given by

$${\Omega }^{•}(U_i)\leftarrow W(\mathrm{Lie}(V//G)):\nabla {\sigma }_i.$$\Omega^\bullet(U_i) \leftarrow W(Lie(V//G)) : \nabla \sigma_i \,.

In degree 0 this is an algebra homomorphism

$$C^{\mathrm{\infty }}(U_i)\leftarrow C^{\mathrm{\infty }}(V):{\sigma }_i.$$C^\infty(U_i) \leftarrow C^\infty(V) : \sigma_i \,.

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

$${\Omega }^1(U_i)\leftarrow {𝔤}^{*}:A_i,$$\Omega^1(U_i) \leftarrow \mathfrak{g}^* : A_i \,,

which is the connection form itself, as well as a linear map

$${\Omega }^1(U_i)\leftarrow {\Omega }^1(V):F_{\nabla {\sigma }_i}^1,$$\Omega^1(U_i) \leftarrow \Omega^1(V) : F^1_{\nabla \sigma_i} \,,

which is the curvature 1-form. The respect of these maps for the differential says that

$$F_{\nabla {\sigma }_i}^1=d_{\mathrm{dR}}{\sigma }_i+\rho (A)({\sigma }_i).$$F^1_{\nabla \sigma_i} = d_{dR} \sigma_i + \rho(A)(\sigma_i) \,.

This is the familiar local formula for a covariant derivative as one finds it in the literature. We therefore write for short

$${\nabla }_{(-)}{\sigma }_i:=F_{\nabla {\mathrm{sigma}}_i}^1$$\nabla_{(-)} \sigma_i := F^1_{\nabla sigma_i}

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

$${\nabla }_{(-)}:\Gamma (P{\times }_{\rho }V)\to \Gamma (P{\times }_{\rho }V).$$\nabla_{(-)} : \Gamma(P \times_\rho V) \to \Gamma(P \times_\rho V) \,.

There is a Bianchi identity on every curvature component, induced from the respect for differentials of the dg-algebra morphism ${\Omega }^{•}(U_i)\leftarrow W(\mathrm{Lie}(V//G)):F_{\nabla }$ 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 ${}_i$)

$$d_{\mathrm{dR}}{\nabla }_{(-)}{\sigma }^k={\rho }_a{}^k{}_k{F}_A^a\wedge {\sigma }^k-{\rho }_a{}^k{}_l{A}^a{d}_{\mathrm{dR}}{\sigma }^k,.$$d_{dR} \nabla_{(-)} \sigma^k = \rho_a{}^k{}_k F_A^a \wedge \sigma^k - \rho_a{}^k{}_l A^a d_{dR} \sigma^k ,.

More invariantly we may write this as

$${\nabla }_{(-)}{\nabla }_{(-)}\sigma =\rho (F_A)(\sigma )$$\nabla_{(-)} \nabla_{(-)} \sigma = \rho(F_A)(\sigma)

and this find the usual expression of the curvature of a connection as the square of the covariant derivative.

Related concepts

gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map

References

A discussion of covariant derivatives for a Levi-Civita connection in terms of synthetic differential geometry is in

• Gonzalo Reyes, Covariant derivation (pdf)