nLab connection on a smooth principal infinity-bundle

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 $(\infty,1)$-topos Smooth∞Grpd of smooth ∞-groupoids.

For $G$ an ∞-Lie group, a connection on a smooth $G$-principal ∞-bundle is a structure that supports the Chern-Weil homomorphism in Smooth∞Grpd: it interpolates between the nonabelian cohomology class $c \in H^1_{smooth}(X,G)$ 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 $\mathbf{H}$ be a cohesive (∞,1)-topos equippd with differential cohesion and let $\mathbb{G} \in Grp(\mathbf{H})$ be a braided ∞-group. Write

$$curv_{\mathbb{G}} = \theta_{\mathbf{B}\mathbb{G}} \;\colon\; \mathbf{B}\mathbb{G} \to \flat_{dR}\mathbf{B}^2 \mathbb{G}$$

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

Let

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

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

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

is a 1-epimorphism.

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

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

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

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

For $\infty$-groups obtained by Lie integration

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 $G$-connections as cocycles in the stack $\mathbf{B}G_{conn}$ as discussed at connection on a bundle, in view of the characterization of Weil algebra in the smooth infinity-topos

Chevalley-Eilenberg algebra CE $\leftarrow$ Weil algebra W $\leftarrow$ invariant polynomials inv

differential forms on moduli stack $\mathbf{B}G_{conn}$ of principal connections (Freed-Hopkins 13):

$$\array{ CE(\mathfrak{g}) &\simeq& \Omega^\bullet_{li \atop cl}(G) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\simeq & \Omega^\bullet(\mathbf{E}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\simeq& \Omega^\bullet(\mathbf{B}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})/G) }$$

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

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

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

Definition

A morphism

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

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

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

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

Its curvature is the composite of morphisms of graded vector spaces

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

that injects the shifted generators into the Weil algebra.

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

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

in which case we call $A$ flat.

The curvature characteristic forms of $A$ are the composite

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

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

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

Definition

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

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

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

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

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

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

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

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

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

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 $f : K \to U$ and $\rho : [k] \to [l]$ is by the evident precomposition with the pullback of differential forms $\Omega^\bullet(U \times \Delta^k) \stackrel{(f,id)^*}{\to} \Omega^\bullet(K \times \Delta^k)$ and $\Omega^\bullet(U \times \Delta^l) \stackrel{(id,\rho)^*}{\leftarrow} \Omega^\bullet(U, \times \Delta^k)$.

Proposition

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

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

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

Proof

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

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

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

1. $\iota_v \langle F_A \rangle = 0$;

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

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

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

$$\mathcal{L}_v \langle F_A \rangle = d \iota_v \langle F_A \rangle + \iota_v d \langle F_A \rangle = 0 \,.$$

Remark

For a general L-∞ algebra $\mathfrak{g}$ the curvature forms $F_A$ 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 $n \in \mathbb{N}$ let $\mathbf{cosk}_{n+1} : sSet \to sSet$ be the simplicial coskeleton functor. Its prolongation to simplicial presheaves we denote here $\tau_n$ and write

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

etc. This is the delooping

$$\tau_n \exp(\mathfrak{g}) = \mathbf{B}G$$

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

Definition

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

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

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

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

Remark

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

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

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:

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

Remark

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

This distinction is important: over objects $X \in$ Smooth∞Grpd that are not smooth manifolds but for instance orbifolds, the genuine $\mathfrak{g}$-connections for general higher $\mathfrak{g}$ 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 $\exp(\mathfrak{g})_{conn}(U)$ may be thought of as gauge transformations between $\mathfrak{g}$-valued forms. We unwind what these look like concretely.

Definition

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

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

consider the unique decomposition

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

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

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

We describe now how this enccodes a gauge transformation

$$\lambda : A_0(s=0) \stackrel{}{\to} A_U(s = 1) \,.$$

Observation

We have

$$\frac{d}{d s} A_U = (d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots) + \iota_{\partial_s} F_A \,,$$

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

Proof

This is the result of applying the contraction $\iota_{\partial s}$ to the defining equation for the curvature $F_A$ of $A$ using the nature of the Weil algebra:

$$F_A = d_{dR} A + [A \wedge A] + [A \wedge A \wedge A] + \cdots$$

and inserting the above decomposition for $A$.

Definition

Define the covariant derivative of the gauge parameter to be

$$\nabla_A \lambda := d \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots \,.$$

In this notation we have

• the general identity

  (1)$$\frac{d}{d s} A_U = \nabla \lambda + (F_A)_s$$

• the horizontality or rheonomy constraint or second Ehresmann condition $\iota_{\partial_s} F_A = 0$, the differential equation

  (2)$$\frac{d}{d s} A_U = \nabla \lambda \,.$$

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

Observation

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

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

Ordinary connections on principal 1-bundles

Proposition

(connections on ordinary bundles)

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

$$\tau_1 \exp(\mathfrak{g})_{conn} \simeq \mathbf{B}G_{conn}$$

betweenn the 1-truncated coefficient object for $\mathfrak{g}$-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 $\Omega^1(-,\mathfrak{g})$.

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

$$\frac{\partial}{\partial s} A = d_U \lambda + [\lambda, A] \,.$$

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

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

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

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

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

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

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

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

Further examples

• For $\mathfrak{g}$ Lie 2-algebra, a $\mathfrak{g}$-valued differential form in the sense described here is precisely an Lie 2-algebra valued form.

• For $n \in \mathbb{N}$, a $b^{n-1}\mathbb{R}$-valued differential form is the same as an ordinary differential $n$-form.

• What is called an “extended soft group manifold” in the literature on the D'Auria-Fre formulation of supergravity is precisely a collection of $\infty$-Lie algebroid valued forms with values in a super $\infty$-Lie algebra such as the

supergravity Lie 3-algebra/supergravity Lie 6-algebra (for 11-dimensional supergravity). The way curvature and Bianchi identity are read off from “extded soft group manifolds” in this literature is – apart from this difference in terminology – precisely what is described above.

Related concepts

• connection on a bundle

• connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe

• connection on a 3-bundle / connection on a bundle 2-gerbe

• connection on an ∞-bundle

  • flat ∞-connection

  • adjusted Weil algebra

• parallel transport, higher parallel transport

• holonomy

higher Atiyah groupoid

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

References

See also the rereferences at nonabelian differential cohomology.

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

• Urs Schreiber, Obstructions to $n$-Bundle Lifts Part II (Oct 2007) [diagram: pdf]

• Hisham Sati, Urs Schreiber, Jim Stasheff, L-∞ algebra connections and applications to String- and Chern-Simons n-transport, in: Quantum Field Theory, Birkhäuser (2009) 303-424 (arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17)

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics, October 2012, Volume 315, Issue 1, pp 169-213 (arXiv:0910.4001, doi:10.1007/s00220-012-1510-3)

The global description was then introduced in

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes – An $\infty$-Lie theoretic construction, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (arXiv:1011.4735, euclid:atmp/1358950853)

A more comprehensive account is in sections 3.9.6, 3.9.7 of

• Urs Schreiber, differential cohomology in a cohesive topos

For further developments see the references at adjusted Weil algebra.

Approach via splitting of higher Atiyah Lie algebroids:

• Severin Bunk, Infinitesimal Higher Symmetries and Higher Connections, talk at CQTS (27 Mar 2024) [slides:pdf]