∞-Lie theory (higher geometry)
curvature characteristic form
A curvature characteristic form is a differential form naturally associated to a Lie algebra-valued 1-form that is a measure for the non-triviality of the curvature of the 1-form.
More generally, there is a notion of curvature characteristic forms of L-∞-algebra-valued differential forms and ∞-Lie algebroid valued differential forms.
For $\mathfrak{g}$ a Lie algebra, $\langle -,-, \cdots, -\rangle$ an invariant polynomial of $n$ arguments on the Lie algebra and $A \in \Omega^1(P,\mathfrak{g})$ a Lie-algebra-valued 1-form with curvature 2-form $F_A = d_{dR} A + [A \wedge A]$, the curvature characteristic form of $A$ with respect to $\langle \cdots \rangle$ is the differential form
This form is always an exact form. The $(2 n -1)$-form trivializing it is called a Chern-Simons form.
Notably if $G$ is a Lie group with Lie algebra $\mathfrak{g}$, $P$ is the total space of a $G$-principal bundle $\pi : P \to X$, and $A \in \Omega^1(P,\mathfrak{g})$ is an Ehresmann connection 1-form on $P$ then by the very definition of the $G$-equivariance of $A$ and the invariance of $\langle \cdots \rangle$ it follows that the curvature form is invariant under the $G$-action on $P$ and is therefore the pullback along $\pi$ of a $2 n$-form $P_n \in \Omega^{2 n}(X)$ down on $X$. This form is in general no longer exaxt, but is always a closed form and hence represent a class in the de Rham cohomology of $X$. This establishes the Weil homomorphism from invariant polynomials to de Rham cohomology
The above description of curvature characteristic forms may be formulated in terms of ∞-Lie theory as follows.
For $P \to X$ a $G$-principal bundle write $T X$, $T P$ and $T_{vert} P$ for the tangent Lie algebroid of $X$, of $P$ and the vertical tangent Lie algebroid of $P$, respectively. Write $inn(\mathfrak{g})$ for the Lie 2-algebra given by the differential crossed module $\mathfrak{g}\stackrel{Id}{\to} \mathfrak{g}$ and finally $\prod_i b^{n_i} \mathbb{R}$ for the L-∞-algebra with one abelian generator for each generating invariant polynomial of $\mathfrak{g}$
From the discussion at invariant polynomial we have a canonical morphism $inn(\mathfrak{g}) \to \prod_i b^{n_i}\mathbb{R}$ that represents the generating invariant polynomials.
Recall that a morphism of ∞-Lie algebroids
is equivalently a closed $n$-form on $X$. The data of an Ehresmann connection on $P$ then induces the following diagram of ∞-Lie algebroids
The single curvature characteristic form of a complex line bundle/$U(1)$-principal bundle is the curvature 2-form itself.
A sum of all curvature characteristic forms of a complex vector bundle/U(n)-principal bundle gives the Chern character of a vector bundle.
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