nLab
covariant derivative

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

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 \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 σ\nabla \sigma of a section may be identified with the 1-form curvature-component of a Lie algebroid-valued connection, and the curvature equation

σ=F σ \nabla \nabla \sigma = F_\nabla \sigma

is the Bianchi identity on its curvature 1-form.

Preliminaries on action Lie algebroid cohomology

Let GG be a Lie group, VV a smooth manifold and ρ:G×VV\rho : G \times V \to V a smooth action. Write V//GV//G for the corresponding action groupoid, itself a Lie groupoid. The Lie algebroid Lie(V//G)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

CE(Lie(V//G))=( C (V) 𝔤 *,d ρ), CE(Lie(V//G)) = (\wedge^\bullet_{C^\infty(V)} \mathfrak{g}^*, d_{\rho}) \,,

where the differential acts on functions fC (V)f \in C^\infty(V) by

d ρ:fρ()() *fC (V)𝔤 *. d_\rho : f \mapsto \rho(-)(-)^* f \in C^\infty(V)\otimes \mathfrak{g}^* \,.

Explicitly, for t𝔤t \in \mathfrak{g} this sends ff to the function (d ρf)(t)(d_\rho f)(t) which is the derivative along tT eGt \in T_e G of the function G×VρVfG \times V \stackrel{\rho}{\to}V \stackrel{f}{\to} \mathbb{R}.

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

d ρ:fρ μ at a kf d_\rho : f \mapsto \rho^\mu{}_a t^a \wedge \partial_k f
d ρ:t a12C a bct bt c. d_\rho : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,.

Specifically for VV a finite dimensional vector space, ρ:G\rho : G a linear action, {v k}\{v^k\} a choice of basis of that vector space and ff a linear function f=f kv kf= f_k v^k , we have that (f k:= kf) dimV(f_k := \partial_k f) \in \mathbb{R}^{dim V} are the components vector of the dual vector given by VV in this basis, and the above gives the matrix multiplication form of the action

d ρ:v kt aρ a k lv l. d_\rho : v^k \mapsto t^a \rho_a{}^k{}_l v^l \,.

Notice for completeness that the equation (d ρ) 2=0(d_\rho)^2 = 0 is equivalent to the Jacobi identity of the Lie bracket and the action property of ρ\rho:

d ρd ρv k=(t at bρ a k rρ b r l12C a bct bt cρ 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(Lie(V//G))=( C ( dimV) (Γ(T * dimV)𝔤 *𝔤 *[1]),d W ρ) 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 ρ:v kρ a k lt av l+d dRv k d_{W_\rho} : v^k \mapsto \rho_a{}^k{}_l t^a \wedge v^l + d_{dR} v^k
d W ρ:t a12C a bct bt 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 ρ:d dRv kρ a k kr av lρ a k lt ad dRv 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 aC a bct br 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\mathbf{B}G of GG with the action groupoid of the trivial action on the point, BG*//G\mathbf{B}G \simeq *//G. On Lie algebroids this morphism is dually the inclusion

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

that is the identity on 𝔤 *\mathfrak{g}^*.

Sections of bundles as groupoid principal bundles

For XX smooth manifold, a GG-principal bundle PXP \to X is given by a cocycle in ∞LieGrpd g:XBGg : X \to \mathbf{B}G.

Proposition

The sections σ\sigma of the corresponding ρ\rho-associated bundle P× ρVP \times_\rho V are in natural bijection with the lifts to a V//GV//G-cocycle

V//G σ X g BG. \array{ && V//G \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G } \,.
Proof

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

( (x,j) (x,i) (x,k))( g ij(x) g jk(x) g ik(x) ). \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//GBGV//G \to \mathbf{B}G is in addition a collection of smooth functions {sigma i:U iV}\{sigma_i : U_i \to V\} such that on all U iU jU_i \cap U_j the equation

σ j=ρ(g ij)(σ i) \sigma_j = \rho(g_{i j})(\sigma_i)
(σ i(x) ρ(g ik(x)) σ k(x)) \left( \array{ \sigma_i(x) &&\stackrel{\rho(g_{i k}(x))}{\to}&& \sigma_k(x) } \right)

is satisfied. This identifies the σ i\sigma_i as precisely the components of a section σ\sigma of P× ρGP \times_\rho G with respect to the local trivialization encoded by gg.

Definition of covariant derivative

Observation

Given a connection \nabla on the GG-principal bundle with cocycle gg, there is a unique connection σ\nabla \sigma on the V//GV//G-groupoid principal bundle that corresponds to a section σ\sigma by the above proposition.

Definition

The covariant derivative of a section σ\sigma is the 1-form component F σ 1F_{\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 iXU_i \hookrightarrow X the connection \nabla is given by a morphism of dg-algebras

Ω (U i)W(𝔤):A i \Omega^\bullet(U_i) \leftarrow W(\mathfrak{g}) : A_i

for W(𝔤)W(\mathfrak{g}) the Weil algebra of 𝔤\mathfrak{g}.

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

Ω (U i)W(Lie(V//G)):σ i. \Omega^\bullet(U_i) \leftarrow W(Lie(V//G)) : \nabla \sigma_i \,.

In degree 0 this is an algebra homomorphism

C (U i)C (V):σ i. C^\infty(U_i) \leftarrow C^\infty(V) : \sigma_i \,.

This is the dual of the local section σ i\sigma_i itself. In the case that VV is a vector space with chosen basis {v k}\{v^k\}, we have the corresponding components (σ i k)(\sigma_i^k) of the local section.

Further, in degree 1 the connection is linear map

Ω 1(U i)𝔤 *:A i, \Omega^1(U_i) \leftarrow \mathfrak{g}^* : A_i \,,

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

Ω 1(U i)Ω 1(V):F σ 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 σ i 1=d dRσ i+ρ(A)(σ 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

()σ i:=F sigma i 1 \nabla_{(-)} \sigma_i := F^1_{\nabla sigma_i}

If we keep \nabla fixed and let σ i\sigma_i vary, then this may be thought of as a 1-form with values in endomorphisms of the space of sections

():Γ(P× ρV)Γ(P× ρV). \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 Ω (U i)W(Lie(V//G)):F \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{}_i)

d dR ()σ k=ρ a k kF A aσ kρ a k lA ad dRσ 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

() ()σ=ρ(F A)(σ) \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.

gauge field: models and components

physicsdifferential geometrydifferential cohomology
gauge fieldconnection on a bundlecocycle in differential cohomology
instanton/charge sectorprincipal bundlecocycle in underlying cohomology
gauge potentiallocal connection differential formlocal connection differential form
field strengthcurvatureunderlying cocycle in de Rham cohomology
gauge transformationequivalencecoboundary
minimal couplingcovariant derivativetwisted cohomology
BRST complexLie algebroid of moduli stackLie algebroid of moduli stack
extended Lagrangianuniversal Chern-Simons n-bundleuniversal characteristic map

References

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

Revised on August 14, 2013 13:21:17 by Urs Schreiber (24.131.18.91)