∞-Lie theory (higher geometry)
Formal Lie groupoids
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 a Lie algebra, an invariant polynomial of arguments on the Lie algebra and a Lie-algebra-valued 1-form with curvature 2-form , the curvature characteristic form of with respect to is the differential form
This form is always an exact form. The -form trivializing it is called a Chern-Simons form.
Notably if is a Lie group with Lie algebra , is the total space of a -principal bundle , and is an Ehresmann connection 1-form on then by the very definition of the -equivariance of and the invariance of it follows that the curvature form is invariant under the -action on and is therefore the pullback along of a -form down on . 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 . This establishes the Weil homomorphism from invariant polynomials to de Rham cohomology
In terms of -Lie algebroids
The above description of curvature characteristic forms may be formulated in terms of ∞-Lie theory as follows.
For a -principal bundle write , and for the tangent Lie algebroid of , of and the vertical tangent Lie algebroid of , respectively. Write for the Lie 2-algebra given by the differential crossed module and finally for the L-∞-algebra with one abelian generator for each generating invariant polynomial of
From the discussion at invariant polynomial we have a canonical morphism that represents the generating invariant polynomials.
Recall that a morphism of ∞-Lie algebroids
is equivalently a closed -form on . The data of an Ehresmann connection on then induces the following diagram of ∞-Lie algebroids