nLab
curvature characteristic form

• ∞-Lie algebroid valued differential forms

• connection on an ∞-bundle

• curvature

• Bianchi identity

• curvature characteristic form

• Chern-Simons form

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Idea

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.

Of connection 1-forms

For $𝔤$ a Lie algebra, $⟨-,-,\cdots ,-⟩$ an invariant polynomial of $n$ arguments on the Lie algebra and $A\in {\Omega }^1(P,𝔤)$ a Lie-algebra-valued 1-form with curvature 2-form $F_A=d_{\mathrm{dR}}A+[A\wedge A]$, the curvature characteristic form of $A$ with respect to $⟨\cdots ⟩$ is the differential form

$$⟨F_A\wedge \cdots \wedge F_A⟩\in {\Omega }^{2n}(P).$$\langle F_A \wedge \cdots \wedge F_A \rangle \in \Omega^{2 n}(P) \,.

This form is always an exact form. The $(2n-1)$-form trivializing it is called a Chern-Simons form.

Notably if $G$ is a Lie group with Lie algebra $𝔤$, $P$ is the total space of a $G$-principal bundle $\pi :P\to X$, and $A\in {\Omega }^1(P,𝔤)$ is an Ehresmann connection 1-form on $P$ then by the very definition of the $G$-equivariance of $A$ and the invariance of $⟨\cdots ⟩$ it follows that the curvature form is invariant under the $G$-action on $P$ and is therefore the pullback along $\pi$ of a $2n$-form $P_n\in {\Omega }^{2n}(X)$ 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

In terms of $\mathrm{\infty }$-Lie algebroids

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 $TX$, $TP$ and $T_{\mathrm{vert}}P$ for the tangent Lie algebroid of $X$, of $P$ and the vertical tangent Lie algebroid of $P$, respectively. Write $\mathrm{inn}(𝔤)$ for the Lie 2-algebra given by the differential crossed module $𝔤\stackrel{\mathrm{Id}}{\to }𝔤$ and finally ${\prod }_i{b}^{n_i}ℝ$ for the L-∞-algebra with one abelian generator for each generating invariant polynomial of $𝔤$

From the discussion at invariant polynomial we have a canonical morphism $\mathrm{inn}(𝔤)\to {\prod }_i{b}^{n_i}ℝ$ that represents the generating invariant polynomials.

Recall that a morphism of ∞-Lie algebroids

$$TX\to b^nℝ$$T X \to b^n \mathbb{R}

is equivalently a closed $n$-form on $X$. The data of an Ehresmann connection on $P$ then induces the following diagram of ∞-Lie algebroids

$$\begin{array}{cccccc}{T}_{\mathrm{vert}}P& \stackrel{A_{\mathrm{vert}}}{\to }& 𝔤& & & \mathrm{flat}\mathrm{vertical}\mathrm{form}\\ \downarrow & & \downarrow & & & \mathrm{first}\mathrm{Ehresmann}\mathrm{condition}\\ TP& \stackrel{A}{\to }& \mathrm{inn}(𝔤)& & & \mathrm{form}\mathrm{on}\mathrm{total}\mathrm{space}\\ \downarrow & & \downarrow & & & \mathrm{second}\mathrm{Ehresmann}\mathrm{condition}\\ TX& \stackrel{(P_n)}{\to }& \prod _i{b}^{n_i}ℝ& & & \mathrm{curvature}\mathrm{characteristic}\mathrm{forms}\end{array}.$$\array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} &&& flat vertical form \\ \downarrow && \downarrow &&& first Ehresmann condition \\ T P &\stackrel{A}{\to}& inn(\mathfrak{g}) &&& form on total space \\ \downarrow && \downarrow &&& second Ehresmann condition \\ T X &\stackrel{(P_n)}{\to}& \prod_i b^{n_i} \mathbb{R} &&& curvature characteristic forms } \,.

Examples

• 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.

References

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