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connection on a smooth principal infinity-bundle

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

In every cohesive (∞,1)-topos there is an intrinsic notion of ∞-Chern-Weil theory that gives rise to a notion of connection on principal ∞-bundles. We describe here details of the realization of this general abstract structure in the cohesive (,1)(\infty,1)-topos Smooth∞Grpd of smooth ∞-groupoids.

For GG an ∞-Lie group, a connection on a smooth GG-principal ∞-bundle is a structure that supports the Chern-Weil homomorphism in Smooth∞Grpd: it interpolates between the nonabelian cohomology class cH smooth 1(X,G)c \in H^1_{smooth}(X,G) of the bundle and the refinements to ordinary differential cohomology of its characteristic classes: the curvature characteristic classes.

This generalizes the notion of connection on a bundle and the ordinary Chern-Weil homomorphism in differential geometry.

See the Motivation section at Chern-Weil theory in Smooth∞Grpd and the page ∞-Chern-Weil theory introduction for more background.

Definition

For braided \infty-groups

Let H\mathbf{H} be a cohesive (∞,1)-topos equippd with differential cohesion and let 𝔾Grp(H)\mathbb{G} \in Grp(\mathbf{H}) be a braided ∞-group. Write

curv 𝔾:θ B𝔾 dRB 2𝔾 curv_{\mathbb{G}} \;\colon\; \theta_{\mathbf{B}\mathbb{G}} \to \flat_{dR}\mathbf{B}^2 \mathbb{G}

for the Mauerr-Cartan form? on the delooping ∞-group B𝔾Grp(H)\mathbf{B}\mathbb{G} \in Grp(\mathbf{H}).

Let

Ω(,𝔾) dRB 2𝔾 \Omega(-,\mathbb{G}) \to \flat_{dR}\mathbf{B}^2 \mathbb{G}

be the morphism out of a 0-truncated object which is universal with the property that for ΣH\Sigma \in \mathbf{H} any manifold, the induced internal hom map

[Σ,Ω(,𝔾)][Σ, dRB 2𝔾] [\Sigma, \Omega(-,\mathbb{G})] \to [\Sigma, \flat_{dR}\mathbf{B}^2 \mathbb{G}]

is a 1-epimorphism.

Then write B𝔾 conn\mathbf{B}\mathbb{G}_{conn} for the (∞,1)-pullback in

B𝔾 conn Ω(,𝔾) B𝔾 curv 𝔾 dRB 2𝔾. \array{ \mathbf{B}\mathbb{G}_{conn} &\to& \Omega(-,\mathbb{G}) \\ \downarrow && \downarrow \\ \mathbf{B}\mathbb{G} &\stackrel{curv_{\mathbb{G}}}{\to}& \flat_{dR}\mathbf{B}^2 \mathbb{G} } \,.

We say that B𝔾 conn\mathbf{B}\mathbb{G}_{conn} is the moduli ∞-stack of 𝔾\mathbb{G}-principal \infty-connections.

For instance for 𝔾=B n1U(1)\mathbb{G} = \mathbf{B}^{n-1}U(1) the circle n-group the moduli nn-stack B nU(1) conn\mathbf{B}^n U(1)_{conn} is presented by the Deligne complex for ordinary differential cohomology in degree (n+1)(n+1), hence is the moduli nn-stack for circle n-bundles with connection.

For \infty-groups obtained by Lie interation

We assume that the reader is familiar with the notation and constructions discussed at Smooth∞Grpd. The following definition may be understood as a direct generalization of the description of ordinary GG-connections as cocycles in the stack BG conn\mathbf{B}G_{conn} as discussed at connection on a bundle.

We discuss now connections on those GG-principal ∞-bundles for which GG \in Smooth∞Grpd is an smooth ∞-group that arises from Lie integration of an L-∞ algebra 𝔤\mathfrak{g}.

Let 𝔤L CE\mathfrak{g} \in L_\infty \stackrel{CE}{\hookrightarrow} dgAlg op{}^{op} be an L-∞ algebra over the real numbers and of finite type with Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) and Weil algebra W(𝔤)W(\mathfrak{g}).

For XX a smooth manifold, write Ω (X)dgAlg\Omega^\bullet(X) \in dgAlg for the de Rham complex of smooth differential forms. For kk \in \mathbb{N} let Δ k\Delta^k be the standard kk-simplex regarded as a smooth manifold with corners in the standard way. Write Ω si (X×Δ k)\Omega^\bullet_{si}(X \times \Delta^k) for the sub-dg-algebra of differential forms with sitting instants perpendicular to the boundary of the simplex, and Ω si,vert (X×Δ k)\Omega^\bullet_{si,vert}(X\times \Delta^k) for the further sub-dg-algebra of vertical differential forms with respect to the canonical projection X×Δ kXX \times \Delta^k \to X.

Definition

A morphism

Ω (X)W(𝔤):A \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A

in dgAlg we call an L-∞ algebra valued differential form with values in 𝔤\mathfrak{g}, dually a morphism of ∞-Lie algebroids

A:TXinn(𝔤) A : T X \to inn(\mathfrak{g})

from the tangent Lie algebroid to the inner automorphism ∞-Lie algebra.

Its curvature is the composite of morphisms of graded vector spaces

Ω (X)AW(𝔤)F ()𝔤 *[2]:F A \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{F_{(-)}}{\leftarrow} \mathfrak{g}^*[2] : F_{A}

that injects the shifted generators into the Weil algebra.

Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebra

(F A=0)( CE(𝔤) A flat Ω (X) A W(𝔤)) (F_A = 0) \;\;\Leftrightarrow \;\; \left( \array{ && CE(\mathfrak{g}) \\ & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right)

in which case we call AA flat.

The curvature characteristic forms of AA are the composite

Ω (X)AW(𝔤)F ()inv(𝔤):F A, \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{\langle F_{(-)} \rangle}{\leftarrow} inv(\mathfrak{g}) : \langle F_A\rangle \,,

where inv(𝔤)W(𝔤)inv(\mathfrak{g}) \to W(\mathfrak{g}) is the inclusion of the invariant polynomials.

We define now simplicial presheaves over the site CartSp smooth{}_{smooth} \hookrightarrow SmoothMfd of Cartesian spaces and smooth functions between them.

Definition

Write exp(𝔤)[CartSp smooth op,sSet]\exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet] for the simplicial presheaf given by

exp(𝔤):(U,[k]){Ω si,vert (U×Δ k)A vertCE(𝔤)} \exp(\mathfrak{g}) : (U,[k]) \mapsto \left\{ \Omega^\bullet_{si,vert}(U \times\Delta^k) \stackrel{A_{vert}}{\leftarrow} CE(\mathfrak{g}) \right\}

(the untruncated Lie integration of 𝔤\mathfrak{g}).

Write exp(𝔤) diff[CartSp smooth op,sSet]\exp(\mathfrak{g})_{diff} \in [CartSp_{smooth}^{op}, sSet] for the simplicial presheaf given by

exp(𝔤) diff:(U,[k]){Ω si,vert (U×Δ k) A vert CE(𝔤) Ω si (U×Δ k) A W(𝔤)}. \exp(\mathfrak{g})_{diff} : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times\Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right\} \,.

Write exp(𝔤) ChW[CartSp smooth op,sSet]\exp(\mathfrak{g})_{ChW} \in [CartSp_{smooth}^{op}, sSet] for the simplicial presheaf given by

exp(𝔤) ChW:(U,[k]){Ω si,vert (U×Δ k) A vert CE(𝔤) Ω si (U×Δ k) A W(𝔤) Ω (U) F A inv(𝔤)}. \exp(\mathfrak{g})_{ChW} : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times\Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A\rangle}{\leftarrow}& inv(\mathfrak{g}) } \right\} \,.

Define the simplicial presheaf exp(𝔤) conn\exp(\mathfrak{g})_{conn} by

exp(𝔤) conn(U):[k]{Ω si (U×Δ k)AW(𝔤)|vΓ(TΔ k):ι vF A=0} \exp(\mathfrak{g})_{conn}(U) : [k] \mapsto \left\{ \Omega^\bullet_{si}(U \times \Delta^k) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \;\; | \;\; \forall v \in \Gamma(T \Delta^k) : \iota_v F_A = 0 \right\}

Here on the right we have in each case the sets of horizontal morphisms in dgAlg that make commuting diagrams in dgAlg as indicated, with the vertical morphisms being the canonical projections and inclusions. The functoriality in f:KUf : K \to U and ρ:[k][l]\rho : [k] \to [l] is by the evident precomposition with the pullback of differential forms Ω (U×Δ k)(f,id) *Ω (K×Δ k)\Omega^\bullet(U \times \Delta^k) \stackrel{(f,id)^*}{\to} \Omega^\bullet(K \times \Delta^k) and Ω (U×Δ l)(id,ρ) *Ω (U,×Δ k)\Omega^\bullet(U \times \Delta^l) \stackrel{(id,\rho)^*}{\leftarrow} \Omega^\bullet(U, \times \Delta^k).

Proposition

There are canonical morphisms in [CartSp smooth op,sSet][CartSp_{smooth}^{op},sSet] between these objects

exp(𝔤) conn exp(𝔤) ChW exp(𝔤) diff exp(𝔤), \array{ \exp(\mathfrak{g})_{conn} &\hookrightarrow& \exp(\mathfrak{g})_{ChW} &\hookrightarrow& \exp(\mathfrak{g})_{diff} \\ && && \downarrow \\ && && \exp(\mathfrak{g}) } \,,

where the horizontal morphisms are monomorphisms of simplicial presheaves and the vertical morphism is over each UCartSpU \in CartSp an equivalence of Kan complexes (it is a weak equivalence between fibrant objects in the projective model structure on simplicial presheaves).

Proof

The inclusion exp(𝔤) ChWexp(𝔤) dff\exp(\mathfrak{g})_{ChW} \hookrightarrow \exp(\mathfrak{g})_{dff} is clear. The weak equivalence exp(𝔤) diffexp(𝔤)\exp(\mathfrak{g})_{diff} \to \exp(\mathfrak{g}) is discussed at Smooth∞Grpd (but is also directly verified).

To see the inclusion exp(𝔤) connexp(𝔤) ChW\exp(\mathfrak{g})_{conn} \hookrightarrow \exp(\mathfrak{g})_{ChW} we need to check that the horizonality condition ι vF A=0\iota_v F_A = 0 on the curvature of a 𝔤\mathfrak{g}-valued form AA for all vector fields vv tangent to the simplex implies that all the curvature characteristic forms F A\langle F_A\rangle are basic forms that “descend to UU”, hence that are in the image of the inclusion Ω (U)Ω si (U×Δ k)\Omega^\bullet(U) \to \Omega^\bullet_{si}(U \times \Delta^k).

For this it is sufficient to show that for all vΓ(TΔ k)v \in \Gamma(T \Delta^k) we have

  1. ι vF A=0\iota_v \langle F_A \rangle = 0;

  2. vF A=0\mathcal{L}_v \langle F_A \rangle = 0

where in the second line we have the Lie derivative v\mathcal{L}_v along vv.

The first condition is evidently satisfied if already ι vF A=0\iota_v F_A = 0. The second condition follows with Cartan calculus and using that d dRF A=0d_{dR} \langle F_A\rangle = 0 (which holds as a consequence of the definition of invariant polynomial):

vF A=dι vF A+ι vdF A=0. \mathcal{L}_v \langle F_A \rangle = d \iota_v \langle F_A \rangle + \iota_v d \langle F_A \rangle = 0 \,.
Remark

For a general L-∞ algebra 𝔤\mathfrak{g} the curvature forms F AF_A themselves are not necessarily closed (rather they satisfy the Bianchi identity), hence requiring them to have no component along the simplex does not imply that they descend. This is different for abelian L-∞ algebras: for them the curvature forms themselves are already closed, and hence are themselves already curvature characteristics that do descent.

For nn \in \mathbb{N} let cosk n+1:sSetsSet\mathbf{cosk}_{n+1} : sSet \to sSet be the simplicial coskeleton functor. Its prolongation to simplicial presheaves we denote here τ n\tau_n and write

τ nexp(𝔤)[CartSp smooth op,sSet] \tau_n \exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet]

etc. This is the delooping

τ nexp(𝔤)=BG \tau_n \exp(\mathfrak{g}) = \mathbf{B}G

of the universal Lie integration of 𝔤\mathfrak{g} to an smooth n-group GG.

Definition

For any X[CartSp smooth op,sSet]X \in [CartSp_{smooth}^{op}, sSet] and X^X\hat X \to X any cofibrant resolution in the local projective model structure on simplicial presheaves (see Smooth∞Grpd for details), we say that the sSet-hom object

  • [CartSp smooth op,sSet](X^,τ nexp(𝔤))[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g})) is the ∞-groupoid of smooth GG-principal ∞-bundles on XX;

  • [CartSp smooth op,sSet](X^,τ nexp(𝔤) diff)[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g})_{diff}) is the ∞-groupoid of smooth GG-principal ∞-bundles on XX equipped with pseudo \infty-connection;

  • [CartSp smooth op,sSet](X^,τ nexp(𝔤) conn)[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g})_{conn}) is the ∞-groupoid of smooth GG-principal ∞-bundles on XX equipped with \infty-connection.

Remark

In view of this definition we may read the above sequence of morpisms of coefficient objects as follows:

exp(𝔤) conn genuineconnections exp(𝔤) ChW pseudoconnectionwithglobalcurvaturecharacteristics exp(𝔤) diff pseudoconnections exp(𝔤) barebundles, \array{ \exp(\mathfrak{g})_{conn} &&& genuine\;connections \\ \downarrow \\ \exp(\mathfrak{g})_{ChW} &&& pseudo-connection\;with\;global\;curvature\;characteristics \\ \downarrow \\ \exp(\mathfrak{g})_{diff} &&& pseudo-connections \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{g}) &&& bare bundles } \,,

As we shall see in more detail below, the components of an \infty-connection in terms of the above diagrams we may think of as follows:

Ω (U×Δ k) vert A vert CE(𝔤) gaugetransformation Ω (U×Δ k) A W(𝔤) 𝔤valuedform Ω (U) F A inv(𝔤) curvaturecharacteristicforms \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& gauge\;transformation \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &&& \mathfrak{g}-valued\;form \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A\rangle}{\leftarrow}& inv(\mathfrak{g}) &&& curvature\;characteristic\;forms }
Remark

In full Chern-Weil theory in Smooth∞Grpd the fundamental object of interest is really exp(𝔤) diff\exp(\mathfrak{g})_{diff} – the object of pseudo-connections, which serves as the correspondence object for an ∞-anafunctor out of exp(𝔤)\exp(\mathfrak{g}) that presents the differential characteristic classes on exp(𝔤)\exp(\mathfrak{g}). From an abstract point of view the other objects only serve the purpose of picking particularly nice representatives.

This distinction is important: over objects XX \in Smooth∞Grpd that are not smooth manifolds but for instance orbifolds, the genuine 𝔤\mathfrak{g}-connections for general higher 𝔤\mathfrak{g} do not exhaust all nonabelian differential cocycles. This just means that not every differential class in this case does have a nice representative.

Examples

1-Morphisms: integration of infinitesimal gauge transformations

The 1-morphisms in exp(𝔤) conn(U)\exp(\mathfrak{g})_{conn}(U) may be thought of as gauge transformations between 𝔤\mathfrak{g}-valued forms. We unwind what these look like concretely.

Definition

Given a 1-morphism in exp(𝔤)(X)\exp(\mathfrak{g})(X), represented by 𝔤\mathfrak{g}-valued forms

Ω (U×Δ 1)W(𝔤):A \Omega^\bullet(U \times \Delta^1) \leftarrow W(\mathfrak{g}) : A

consider the unique decomposition

A=A U+(A vert:=λds), A = A_U + ( A_{vert} := \lambda \wedge d s) \; \; \,,

with A UA_U the horizonal differential form component and s:Δ 1=[0,1]s : \Delta^1 = [0,1] \to \mathbb{R} the canonical coordinate.

We call λ\lambda the gauge parameter . This is a function on Δ 1\Delta^1 with values in 0-forms on UU for 𝔤\mathfrak{g} an ordinary Lie algebra, plus 1-forms on UU for 𝔤\mathfrak{g} a Lie 2-algebra, plus 2-forms for a Lie 3-algebra, and so forth.

We describe now how this enccodes a gauge transformation

λ:A 0(s=0)A U(s=1). \lambda : A_0(s=0) \stackrel{}{\to} A_U(s = 1) \,.
Observation

We have

ddsA U=(d Uλ+[λA]+[λAA]+)+ι sF A, \frac{d}{d s} A_U = (d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots) + \iota_{\partial_s} F_A \,,

where the sum is over all higher brackets of the L-∞ algebra 𝔤\mathfrak{g}.

Proof

This is the result of applying the contraction ι s\iota_{\partial s} to the defining equation for the curvature F AF_A of AA using the nature of the Weil algebra:

F A=d dRA+[AA]+[AAA]+ F_A = d_{dR} A + [A \wedge A] + [A \wedge A \wedge A] + \cdots

and inserting the above decomposition for AA.

Definition

Define the covariant derivative of the gauge parameter to be

Aλ:=dλ+[Aλ]+[AAλ]+. \nabla_A \lambda := d \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots \,.

In this notation we have

  • the general identity

    (1)ddsA U=λ+(F A) s \frac{d}{d s} A_U = \nabla \lambda + (F_A)_s
  • the horizontality or rheonomy constraint or second Ehresmann condition ι sF A=0\iota_{\partial_s} F_A = 0, the differential equation

    (2)ddsA U=λ. \frac{d}{d s} A_U = \nabla \lambda \,.

This is known as the equation for infinitesimal gauge transformations of an L L_\infty-algebra valued form.

Observation

By Lie integration we have that A vertA_{vert} – and hence λ\lambda – defines an element exp(λ)\exp(\lambda) in the ∞-Lie group that integrates 𝔤\mathfrak{g}.

The unique solution A U(s=1)A_U(s = 1) of the above differential equation at s=1s = 1 for the initial values A U(s=0)A_U(s = 0) we may think of as the result of acting on A U(0)A_U(0) with the gauge transformation exp(λ)\exp(\lambda).

Ordinary connections on principal 1-bundles

Proposition

(connections on ordinary bundles)

For 𝔤\mathfrak{g} an ordinary Lie algebra with simply connected Lie group GG and for BG conn\mathbf{B}G_{conn} the groupoid of Lie algebra-valued forms we have an equivalence

τ 1exp(𝔤) connBG conn \tau_1 \exp(\mathfrak{g})_{conn} \simeq \mathbf{B}G_{conn}

betweenn the 1-truncated coefficient object for 𝔤\mathfrak{g}-valued \infty-connections and the coefficient objects for ordinary connections on a bundle (see there).

Proof

Notice that the sheaves of objects on both sides are manifestly isomorphic, both are the sheaf of Ω 1(,𝔤)\Omega^1(-,\mathfrak{g}).

On morphisms, we have by the above for a form Ω (U×Δ 1)W(𝔤):A\Omega^\bullet(U \times \Delta^1) \leftarrow W(\mathfrak{g}) : A decomposed into a horizontal and a verical pice as A=A U+λdtA = A_U + \lambda \wedge d t that the condition ι tF A=0\iota_{\partial_t} F_A = 0 is equivalent to the differential equation

sA=d Uλ+[λ,A]. \frac{\partial}{\partial s} A = d_U \lambda + [\lambda, A] \,.

For any initial value A(0)A(0) this has the unique solution

A(t) =g(t) 1(A+d U)g(t) =Ad g(t)(A)+g(t) *θ \begin{aligned} A(t) & = g(t)^{-1} (A + d_{U}) g(t) \\ & = Ad_{g(t)}(A) + g(t)^* \theta \end{aligned}

(with θ\theta the Maurer-Cartan form on GG), where gC ([0,1],G)g \in C^\infty([0,1], G) is the parallel transport of λ\lambda:

s(g (t) 1(A+d U)g(t)) =g(t) 1(A+d U)λg(t)g(t) 1λ(A+d U)g(t) \begin{aligned} & \frac{\partial}{\partial s} \left( g_(t)^{-1} (A + d_{U}) g(t) \right) \\ & = g(t)^{-1} (A + d_{U}) \lambda g(t) - g(t)^{-1} \lambda (A + d_{U}) g(t) \end{aligned}

(where for ease of notaton we write actions as if GG were a matrix Lie group).

This implies that the endpoints of the path of 𝔤\mathfrak{g}-valued 1-forms are related by the usual cocycle condition in BG conn\mathbf{B}G_{conn}

A(1)=g(1) 1(A+d U)g(1). A(1) = g(1)^{-1} (A + d_U) g(1) \,.

In the same fashion one sees that given 2-cell in exp(𝔤)(U)\exp(\mathfrak{g})(U) and any 1-form on UU at one vertex, there is a unique lift to a 2-cell in exp(𝔤) conn\exp(\mathfrak{g})_{conn}, obtained by parallel transporting the form around. The claim then follows from the previous statement of Lie integration that τ 1exp(𝔤)=BG\tau_1 \exp(\mathfrak{g}) = \mathbf{B}G.

Further examples

higher Atiyah groupoid

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid version of quantomorphism n-group
coefficient for cohomology:B𝔾\mathbf{B}\mathbb{G}B(B𝔾 conn)\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})B𝔾 conn\mathbf{B} \mathbb{G}_{conn}
type of fiber ∞-bundle:principal ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

References

The local differential form data of \infty-connections was introduced in

The global description was then introduced in

A more comprehensive account is in sections 3.9.6, 3.9.7 of

Revised on July 27, 2014 01:36:55 by Urs Schreiber (89.204.153.39)