This is a sub-entry of ∞-Chern-Simons theory . See there for background and details.
We describe some details of the construction of the ∞-Chern-Weil homomorphism induced by an invariant polynomial $\langle -,-\rangle \in W(\mathfrak{a})$ on an ∞-Lie algebroid $\mathfrak{a}$. As discussed at Chern-Weil homomorphism and ∞-connections, this model for the $\infty$-Chern-Weil homomorphism;
is controlled by a Chern-Simons element $cs \in W(\mathfrak{a})$ that exhibits the transgression of $\langle \rangle$ to an invariant polynomial $\mu \in CE(\mathfrak{a})$;
sends an $\mathfrak{a}$-connection on an ∞-bundle given locally by an ∞-Lie algebroid valued differential form $A \in \Omega^1(U,\mathfrak{a})$ to the circle n-bundle with connection whose local connection $n$-form is Chern-Simons form $cs(A)$.
The Lagrangian of $\infty$-Chern-Simons theory is then of the form $A \mapsto \int_\Sigma cs(A)$. Therefore the identification of $\infty$-Chern-Simons theories essentially reduces to the identificaton of higher Chern-Simons elements $cs$ on ∞-Lie algebroids. In the following we survey the relevant theory for this construction.
Abstractly, an ∞-Lie algebroid is a synthetic-differential ∞-groupoid all whose k-morphisms are infinitesimal, for all $k \geq 1$. Concretely, these objects may be defined as formal duals of certain cosimplicial algebras and equivalrently but more conveniently, using the monoidal Dold-Kan correspondence, of graded-commutative semifree dg-algebras. This perspective we describe now.
We fix a commutative $\mathbb{R}$-algebra $R$. If $L$ is graded $R$-module, then the suspension of $L$ is denoted as $L[-1]$ where $L[-1]_{i}=L_{i-1}$. We say $L$ has finite type if it is degree-wise finitely generated. Given a chain complex $(L,d)$ of finite type, its dual $L^{\ast}$ is the cochain complex $Hom_{R}^{-\bullet}(L,R)$. Hence if $L$ is concentrated in non-negative degrees, so is $L^{\ast}.$
Let $(L,d)$ be a chain complex of $R$-modules of finite type concentrated in non-negative degrees. A Lie $\infty$-algebroid structure on $L$ is a degree 1 $R$-derivation $d_{CE}$ on the graded commutative algebra $CE(L):=S^{\bullet}_{R}(L^{\ast}[-1])$ such that:
Such a chain complex endowed with a Lie $\infty$-algebroid structure is called Lie $\infty$-algebroid. If $L$ is a Lie $\infty$-algebroid, and the underlying complex is concentrated in degrees $0,\dots n-1$, then we say $L$ is a Lie $n$-algebroid.
In analogy with the case in Lie theory, if $L$ is a Lie $\infty$-algebroid, we call the DGCA $(CE(L),d_{CE})$ the Chevalley-Eilenberg algebra of $L$. Note that $CE(L)$ is in fact a quasi-free DGCA i.e. it is free as a graded commutative algebra. A morphism of Lie $\infty$-algebroids $L_1 \to L_2$ is a dgca morphism $CE(L_2) \to CE(L_{1})$.
(want to expand this example out explicitly later)
In differential geometry, a Lie algebroid is traditionally thought of as a vector bundle $E \to M$ equipped with a bundle morphism $\rho \colon E \to T M$ to the tangent bundle and a Lie bracket on the space of sections satisfying some compatibility axioms. It is well-known that a Lie algebroid structure gives a degree 1 differential on the dgca $S^{\bullet}_{R}(L^{\ast}[-1])$, where $R=C^{\infty}(M)$ and $L= \Gamma(E)$. Hence a Lie algebroid is a Lie $\infty$-algebroid in the sense of def. 1. Similarly, Courant algebroids are also examples of Lie $\infty$-algebroids (Roytenberg99).
In general, there is a simple geometric interpretation of our Lie $\infty$-algebroids. Given a Lie $\infty$-algebroid $L$, the degree 0 part of $CE(L)$ is a commutative $\mathbb{R}$-algebra, which we interpret as the formal dual to the space over which the algebroid is defined. If this commutative algebra is $\mathbb{R}$ itself, then we have a Lie $\infty$-algebroid over the point. It is easy to see this implies that $L$ is an $L_{\infty}$-algebra.
The geometry behind our Lie ${\infty}$-algebroids can be made more explicit by introducing the notion of a cosimplicial smooth algebra…..
In this section, we introduce the $\infty$-analogues of the machinery found in classical Chern-Weil theory which we will then use to construct Chern-Simons elements for Lie $\infty$-algebroids.
The Weil algebra of a Lie $\infty$-algebroid $L$ is the dg-algebra
where the differential $d_W$ is defined to be
We may write this as
where $\mathbf{d}$ is the shift operator and $[d_{CE},\mathbf{d}] = 0$.
The Weil algebra $W(L)$ is, in fact, the (quasi-free DGCA) mapping cone of the identity $id \colon CE(L) \to CE(L)$. Note that when $L$ is a Lie algebra, we recover the usual Weil algebra. It is straightforward to show that $W(L)$ is isomorphic to a free differential algebra and therefore is contractible (c.f. Prop. 6 in SSS09). Hence we have:
If $L$ is a Lie $\infty$-algebroid, then the cochain cohomology of $W(L)$ is trivial.
We also adopt the familiar notion of a cocycle in the Chevalley-Eilenberg algebra:
An $n$-cocycle on a Lie $\infty$-algebroid $L$ is a degree $n$ element $\mu \in CE(L)$ such that $d_{CE}\mu=0$.
This next definition generalizes $ad$-invariant polynomials from Lie theory to ∞-Lie theory:
An invariant polynomial on a Lie $\infty$-algebroid $L$ is an element
such that
and
and hance
The space of invariant polynomials is denoted $inv(L)$.
Clearly $inv(L) \subseteq W(L)$ is a dg-sub algebra with the trivial differential. Note that if $L$ is a Lie algebra, then the $d_W$-closed condition is equivalent to the usual $ad$-invariance.
Now we are ready to define $\infty$-Chern-Simons elements. Recall that there is a canonical surjection $\pi \colon W(L)\twoheadrightarrow CE(L)$ which restricts to the identity on $S^{\bullet}(L^{\ast}[-1])$.
Let $\mu$ be a cocycle on a Lie $\infty$-algebroid $L$ and $p \in inv(L)$ an invariant polynomial. A Chern-Simons element for $\mu$ and $p$ is an element $cs \in W(L)$ such that
and
If a Chern-Simons element exists for $\mu$ and $p$, then, in analogy with the classical case for fiber bundles, we say $p$ is in transgression with $\mu$. Note that we have a short exact sequence of complexes:
Since $inv(L) \subseteq ker \pi$, the properties that a Chern-Simons element must satisfy can be understood in terms of the connecting homomorphism $\H^{n}(CE(L)) \to H^{n+1}(ker \pi))$ of the corresponding long exact sequence in cohomology. In particular, if $\mu \in CE^{n}(L)$ and $p \in inv^{n+1}(L)$, then $cs \in W^{n}(L)$ is a Chern-Simons element iff we have
A Chern-Simons element $cs$ witnessing the transgression from $\langle - \rangle$ to $\mu$ is equivalently a commuting diagram of the form
in dgAlg. On the other hand, an n-connection with values in a Lie $n$-algebroid $\mathfrak{a}$ is a cocycle
with coefficients in the simplicial presheaf $\mathbf{cosk}_{n+1} \exp(\mathfrak{a})_{conn}$ that sends $U \in$ CartSp to the $(n+1)$-coskeleton of the simplicial set, which in degree $k$ is the set of commuting diagrams
such that the curvature forms $F_A$ of the ∞-Lie algebroid valued differential forms $A$ on $U \times \Delta^k$ with values in $\mathfrak{a}$ in the middle are horizontal.
If $\mu$ is an ∞-Lie algebroid cocycle of degree $n$, then the ∞-Chern-Weil homomorphism operates by sending an $\infty$-connection given by a Cech cocycle with values in simplicial sets of such commuting diagrams to the obvious pasting composite
Under the map to the coskeleton the group of such cocycles for line $n$-bundle with connection is quotiented by the discrete group $\Gamma$ of periods of $\mu$, such that the $\infty$-Chern-Weil homomorphism is given by sending the $\infty$-connections presented by (10) to
This is a cocycle for a circle n-bundle with connection, whose connection $n$-form is locally given by the Chern-Simons form $cs(A)$. This is the Lagrangian of the $\infty$-Chern-Simons theory defined by $(\mathfrak{a},\langle - \rangle)$ and evaluated on the given $\infty$-connection. If $\Sigma$ is a smooth manifold of dimension $n$, then the higher holonomy of this circle $n$-bundle over $\Sigma$ is the value of the Chern-Simons action. After a suitable gauge transformation this is given by the integral
This integration process we discuss in more detail in the following section.
The notion of Chern-Simons elements for $L_\infty$-algebras and the associated $\infty$-Chern-Simons Lagrangians is due to
The induced construction of the ∞-Chern-Weil homomorphism with special attention to the Chern-Simons circle 3-bundle and the Chern-Simons circle 7-bundle is in
In the general context of cohesive (∞,1)-toposes $\infty$-Chern-Simons theory is discussed in section 4.3 of