nLab
Bianchi identity

Idea

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

It can be thought of as generalizing the equation $d(dA)=0$ 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 $F_a=dA$. The Bianchi identity in this case is the equation

$$dF=0.$$d F = 0 \,.

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

$$d{F}_A+[A\wedge F_A]=0$$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 $𝔤$ a Lie algebra we have naturally associated two dg-algebras: the Chevalley-Eilenberg algebra $\mathrm{CE}(𝔤)$ and the Weil algebra $W(𝔤)$.

The dg-algebra morphisms

$${\Omega }^{•}(U)leftaarrowW(𝔤):(A,F_A)$$\Omega^\bullet(U) \leftaarrow 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

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

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

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

This identifies the 1-form $A\in {\Omega }^1(U,𝔤)$. This extends uniquely to a morphism of dg-algebras and thereby fixes the image of the shifted generators

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

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

and

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

The image of $t^a$ under ${\Omega }^{•}(U)\leftarrow W(𝔤):(A,F_A)$ is the component $A^a$. The image of $r^a$ is therefore, by respect for the differential on $t^a$

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

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

This is the Bianchi identity.

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

Let now $𝔤$ be an arbitrary ∞-Lie-algebra and $W(𝔤)$ its Weil algebra. Then a collection of ∞-Lie algebra valued differential forms is a dg-algebra morphism

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

It curvature is the composite of morphism of graded vector space

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

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

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

Related concepts

• ∞-Lie algebroid valued differential forms

• connection on an ∞-bundle

• curvature

• Bianchi identity

• curvature characteristic form

• Chern-Simons form

References

The Bianchi identity for ∞-Lie algebroid valued differential forms is discussed in

• SSSI (web)