Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
The Bianchi identity is a differential equation satisfied by curvature data.
It can be thought of as generalizing the equation for a real-valued 1-form to higher degree and nonabelian forms.
Generally it applies to the curvature of ∞-Lie algebroid valued differential forms.
For 2-form curvatures
Let be a smooth manifold.
For a differential 1-form, its curvature 2-form is the de Rham differential . The Bianchi identity in this case is the equation
More generally, for an arbitrary Lie algebra and a Lie-algebra valued 1-form, its curvature is the 2-form . The Bianchi identity in this case is the equation
satisfied by these curvature 2-forms.
Reformulation in terms of Weil algebras
We may reformulate the above identities as follows.
For a Lie algebra we have naturally associated two dg-algebras: the Chevalley-Eilenberg algebra and the Weil algebra .
The dg-algebra morphisms
are precisely in bijection with Lie-algebra valued 1-forms as follows: the Weil algebra is of the form
with one copy of in degree 1, the other in degree 2. By the free nature of the Weil algebra, dg-algebra morphisms are in bijection to their underlying morphisms of vector spaces of generators
This identifies the 1-form . This extends uniquely to a morphism of dg-algebras and thereby fixes the image of the shifted generators
The Bianchi identity is precisely the statement that these linear maps, extended to morphisms of graded algebra, are compatible with the differentials and hence do constitute dg-algebra morphisms.
Concretely, if is a dual basis for and the corresponding dual basis for and the structure constants of the Lie bracket on , then the differential on the Weil algebra is defined on generators by
The image of under is the component . The image of is therefore, by respect for the differential on
Respect for the differential on then implies
This is the Bianchi identity.
For curvature of -Lie algebra valued forms.
Let now be an arbitrary ∞-Lie-algebra and its Weil algebra. Then a collection of ∞-Lie algebra valued differential forms is a dg-algebra morphism
It curvature is the composite of morphism of graded vector space
Since is a homomorphism of dg-algebras, this satisfies
This identity is the Bianchi identity for -Lie algebra valued forms.
The Bianchi identity for ∞-Lie algebroid valued differential forms is discussed in