nLab Bianchi identity



\infty-Chern-Weil theory

Differential cohomology



The Bianchi identity is a differential equation satisfied by curvature data.

It can be thought of as generalizing the equation d(dA)=0d (d A) = 0 for a real-valued 1-form AA to higher degree and nonabelian forms.

Generally it applies to the curvature of ∞-Lie algebroid valued differential forms.


For 2-form curvatures

Let UU be a smooth manifold.

For AΩ 1(U)A \in \Omega^1(U) a differential 1-form, its curvature 2-form is the de Rham differential F A=dAF_A = d A. The Bianchi identity in this case is the equation

dF=0. d F = 0 \,.

More generally, for 𝔤\mathfrak{g} an arbitrary Lie algebra and AΩ 1(U,𝔤)A \in \Omega^1(U,\mathfrak{g}) a Lie-algebra valued 1-form, its curvature is the 2-form F A=dA+[AA]F_A = d A + [A \wedge A]. The Bianchi identity in this case is the equation

dF A+[AF A]=0 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 𝔤\mathfrak{g} a Lie algebra we have naturally associated two dg-algebras: the Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) and the Weil algebra W(𝔤)W(\mathfrak{g}).

The dg-algebra morphisms

Ω (U)W(𝔤):(A,F A) \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(𝔤)= (𝔤 *𝔤 *[1]),d W(𝔤) W(\mathfrak{g}) = \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W(\mathfrak{g})}

with one copy of 𝔤 *\mathfrak{g}^* in degree 1, the other in degree 2. By the free nature of the Weil algebra, dg-algebra morphisms Ω (U)W(𝔤)\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) are in bijection to their underlying morphisms of vector spaces of generators

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

This identifies the 1-form AΩ 1(U,𝔤)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

Ω 2(U)𝔤 *[1]:F A. \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}\{t^a\} is a dual basis for 𝔤 *\mathfrak{g}^* and {r a}\{r^a\} the corresponding dual basis for 𝔤 *[1]\mathfrak{g}^*[1] and {C a bc}\{C^a{}_{b c}\} the structure constants of the Lie bracket [,][-,-] on 𝔤\mathfrak{g}, then the differential d W(𝔤)d_{W(\mathfrak{g})} on the Weil algebra is defined on generators by

d W(𝔤)t a=12C a bct bt c+r a d_{W(\mathfrak{g})} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a


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

The image of t at^a under Ω (U)W(𝔤):(A,F A)\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,F_A) is the component A aA^a. The image of r ar^a is therefore, by respect for the differential on t at^a

r a(F A) a:=dA a+12C a bcA bA c. 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 ar^a then implies

d(F A) a+C a bcA a(F A) c=0. 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 𝔤\mathfrak{g} be an arbitrary ∞-Lie-algebra and W(𝔤)W(\mathfrak{g}) its Weil algebra. Then a collection of ∞-Lie algebra valued differential forms is a dg-algebra morphism

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

It curvature is the composite of morphism of graded vector space

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

Since AA is a homomorphism of dg-algebras, this satisfies

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

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


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:

Generalization to supergeometry (motivated by supergravity):

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.