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.
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
for the Maurer-Cartan form on the delooping ∞-group $\mathbf{B}\mathbb{G} \in Grp(\mathbf{H})$.
Let
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
is a 1-epimorphism.
Then write $\mathbf{B}\mathbb{G}_{conn}$ for the (∞,1)-pullback in
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.
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):
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 $X \times \Delta^k \to X$.
A morphism
in dgAlg we call an L-∞ algebra valued differential form with values in $\mathfrak{g}$, dually a morphism of ∞-Lie algebroids
from the tangent Lie algebroid to the inner automorphism ∞-Lie algebra.
Its curvature is the composite of morphisms of graded vector spaces
that injects the shifted generators into the Weil algebra.
Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebra
in which case we call $A$ flat.
The curvature characteristic forms of $A$ are the composite
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.
Write $\exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet]$ for the simplicial presheaf given by
(the untruncated Lie integration of $\mathfrak{g}$).
Write $\exp(\mathfrak{g})_{diff} \in [CartSp_{smooth}^{op}, sSet]$ for the simplicial presheaf given by
Write $\exp(\mathfrak{g})_{ChW} \in [CartSp_{smooth}^{op}, sSet]$ for the simplicial presheaf given by
Define the simplicial presheaf $\exp(\mathfrak{g})_{conn}$ by
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 $\Omega^\bullet(U \times \Delta^l) \stackrel{(id,\rho)^*}{\leftarrow} \Omega^\bullet(U, \times \Delta^k)$.
There are canonical morphisms in $[CartSp_{smooth}^{op},sSet]$ between these objects
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).
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
$\iota_v \langle F_A \rangle = 0$;
$\mathcal{L}_v \langle F_A \rangle = 0$
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):
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
etc. This is the delooping
of the universal Lie integration of $\mathfrak{g}$ to an smooth n-group $G$.
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.
In view of this definition we may read the above sequence of morpisms of coefficient objects as follows:
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:
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.
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.
Given a 1-morphism in $\exp(\mathfrak{g})(X)$, represented by $\mathfrak{g}$-valued forms
consider the unique decomposition
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
We have
where the sum is over all higher brackets of the L-∞ algebra $\mathfrak{g}$.
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:
and inserting the above decomposition for $A$.
Define the covariant derivative of the gauge parameter to be
In this notation we have
the general identity
the horizontality or rheonomy constraint or second Ehresmann condition $\iota_{\partial_s} F_A = 0$, the differential equation
This is known as the equation for infinitesimal gauge transformations of an $L_\infty$-algebra valued form.
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 $\exp(\lambda)$.
(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
betweenn the 1-truncated coefficient object for $\mathfrak{g}$-valued $\infty$-connections and the coefficient objects for ordinary connections on a bundle (see there).
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
For any initial value $A(0)$ this has the unique solution
(with $\theta$ the Maurer-Cartan form on $G$), where $g \in C^\infty([0,1], G)$ is the parallel transport of $\lambda$:
(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 $\mathbf{B}G_{conn}$
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 $\tau_1 \exp(\mathfrak{g}) = \mathbf{B}G$.
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.
connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe
connection on an ∞-bundle
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 |
The local differential form data of $\infty$-connections was introduced in
Hisham Sati, Urs Schreiber, Jim Stasheff, $L_\infty$-algebra connections in Fauser (eds.) Recent Developments in QFT, Birkhäuser (web)
Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted differential String- and Fivebrane structures (web).
The global description was then introduced in
A more comprehensive account is in sections 3.9.6, 3.9.7 of