nLab Bianchi identity

Idea

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.

Definition

For 2-form curvatures

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

$$d F = 0 \,.$$

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

$$d F_A + [A\wedge F_A] = 0$$

satisfied by these curvature 2-forms.

Reformulation in terms of Weil algebras

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

$$\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,F_A)$$

are precisely in bijection with Lie-algebra valued 1-forms as follows: the Weil algebra is of the form

$$W(\mathfrak{g}) = \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W(\mathfrak{g})}$$

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

$$\Omega^1(U) \leftarrow \mathfrak{g}^* : A \,.$$

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

$$\Omega^2(U) \leftarrow \mathfrak{g}^*[1] : F_A \,.$$

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

$$d_{W(\mathfrak{g})} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$$

and

$$d_{W(\mathfrak{g})} r^a = C^a{}_{b c} t^b \wedge r^c \,.$$

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 $t^a$

$$r^a \mapsto (F_A)^a := d A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c \,.$$

Respect for the differential on $r^a$ then implies

$$d (F_A)^a + C^a{}_{b c} A^a \wedge (F_A)^c = 0 \,.$$

This is the Bianchi identity.

For curvature of $\infty$-Lie algebra valued forms.

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

$$\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : A ,.$$

It curvature is the composite of morphism of graded vector space

$$\Omega^\bullet(U) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{F_{(-)}}{\leftarrow} \mathfrak{g}^*[1] : F_A \,.$$

Since $A$ is a homomorphism of dg-algebras, this satisfies

$$d_{dR} F_A + A(d_{W(\mathfrak{g})}(-)) = 0 \,.$$

This identity is the Bianchi identity for $\infty$-Lie algebra valued forms.

Related concepts

• Gauss law

• ∞-Lie algebroid valued differential forms

• connection on an ∞-bundle

• curvature

• adjusted Weil algebra

• curvature characteristic form

• Chern-Simons form

• Dragon's theorem

References

Named after Luigi Bianchi.

Cartan structural equations and Bianchi identities

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:

• Élie Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Annales scientifiques de l’École Normale Supérieure, Sér. 3, 40 (1923) 325-412 [doi:ASENS_1923_3_40__325_0]

Historical review:

• Erhard Scholz, §2 in: E. Cartan’s attempt at bridge-building between Einstein and the Cosserats – or how translational curvature became to be known as “torsion”, The European Physics Journal H 44 (2019) 47-75 [doi:10.1140/epjh/e2018-90059-x]

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]

• Sigurdur Helgason, §I.8 in: Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]

• 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]

• Cadabra, Cartan structural equations and Bianchi identity

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:

• Hisham Sati, Urs Schreiber, Jim Stasheff, p. 52 in:: $L_\infty$ algebra connections and applications to String- and Chern-Simons n-transport, in: Quantum Field Theory, Birkhäuser (2009) 303-424 [arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17]