The Bianchi identity is a differential equation satisfied by curvature data.
It can be thought of as generalizing the equation $d (d A) = 0$ for a real-valued 1-form $A$ to higher degree and nonabelian forms.
Generally it applies to the curvature of ∞-Lie algebroid valued differential forms.
Let $U$ be a smooth manifold.
For $A \in \Omega^1(U)$ a differential 1-form, its curvature 2-form is the de Rham differential $F_A = d A$. The Bianchi identity in this case is the equation
More generally, for $\mathfrak{g}$ an arbitrary Lie algebra and $A \in \Omega^1(U,\mathfrak{g})$ a Lie-algebra valued 1-form, its curvature is the 2-form $F_A = d A + [A \wedge A]$. The Bianchi identity in this case is the equation
satisfied by these curvature 2-forms.
We may reformulate the above identities as follows.
For $\mathfrak{g}$ a Lie algebra we have naturally associated two dg-algebras: the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ and the Weil algebra $W(\mathfrak{g})$.
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 $\mathfrak{g}^*$ in degree 1, the other in degree 2. By the free nature of the Weil algebra, dg-algebra morphisms $\Omega^\bullet(U) \leftarrow W(\mathfrak{g})$ are in bijection to their underlying morphisms of vector spaces of generators
This identifies the 1-form $A \in \Omega^1(U,\mathfrak{g})$. 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 $\{t^a\}$ is a dual basis for $\mathfrak{g}^*$ and $\{r^a\}$ the corresponding dual basis for $\mathfrak{g}^*[1]$ and $\{C^a{}_{b c}\}$ the structure constants of the Lie bracket $[-,-]$ on $\mathfrak{g}$, then the differential $d_{W(\mathfrak{g})}$ on the Weil algebra is defined on generators by
and
The image of $t^a$ under $\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,F_A)$ is the component $A^a$. The image of $r^a$ is therefore, by respect for the differential on $t^a$
Respect for the differential on $r^a$ then implies
This is the Bianchi identity.
Let now $\mathfrak{g}$ be an arbitrary ∞-Lie-algebra and $W(\mathfrak{g})$ 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 $A$ is a homomorphism of dg-algebras, this satisfies
This identity is the Bianchi identity for $\infty$-Lie algebra valued forms.
Named after Luigi Bianchi.
On Cartan structural equations and their Bianchi identities for curvature and torsion of Cartan moving frames and (Cartan-)connections on tangent bundles (especially in first-order formulation of gravity):
The original account:
Historical review:
Further discussion:
Shiing-Shen Chern, p. 748 of: A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals of Mathematics, Second Series, 45 4 (1944) 747-752 [doi:10.2307/1969302]
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §I.2 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.2: pdf]
C. C. Briggs, A Sequence of Generalizations of Cartan’s Conservation of Torsion Theorem [arXiv:gr-qc/9908034]
Loring Tu, §22 in: Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer (2017) [ISBN:978-3-319-55082-4]
Thoan Do, Geoff Prince, An intrinsic and exterior form of the Bianchi identities, International Journal of Geometric Methods in Modern Physics 14 01 (2017) 1750001 [doi:10.1142/S0219887817500013, arXiv:1501.01123]
Ivo Terek Couto, Cartan Formalism and some computations [pdf, pdf]
Generalization to supergeometry (motivated by supergravity):
Julius Wess, Bruno Zumino, p. 362 of: Superspace formulation of supergravity, Phys. Lett. B 66 (1977) 361-364 [doi:10.1016/0370-2693(77)90015-6]
Richard Grimm, Julius Wess, Bruno Zumino, §2 in: A complete solution of the Bianchi identities in superspace with supergravity constraints, Nuclear Phys. B 152 (1979) 255-265 [doi:10.1016/0550-3213(79)90102-0]
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §III.3.2 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch III.3: pdf]
On Bianchi identities in the generalization of ∞-Lie algebroid valued differential forms:
Last revised on March 17, 2024 at 11:12:02. See the history of this page for a list of all contributions to it.