# nLab Bianchi identity

### Context

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

## Idea

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

It can be thought of as generalizing the equation $d\left(dA\right)=0$ 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 $A\in {\Omega }^{1}\left(U\right)$ 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

$dF=0\phantom{\rule{thinmathspace}{0ex}}.$d F = 0 \,.

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

$d{F}_{A}+\left[A\wedge {F}_{A}\right]=0$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 $\mathrm{CE}\left(𝔤\right)$ and the Weil algebra $W\left(𝔤\right)$.

The dg-algebra morphisms

${\Omega }^{•}\left(U\right)leftaarrowW\left(𝔤\right):\left(A,{F}_{A}\right)$\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

$W\left(𝔤\right)={\wedge }^{•}\left({𝔤}^{*}\oplus {𝔤}^{*}\left[1\right]\right),{d}_{W\left(𝔤\right)}$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 ${\Omega }^{•}\left(U\right)←W\left(𝔤\right)$ are in bijection to their underlying morphisms of vector spaces of generators

${\Omega }^{1}\left(U\right)←{𝔤}^{*}:A\phantom{\rule{thinmathspace}{0ex}}.$\Omega^1(U) \leftarrow \mathfrak{g}^* : A \,.

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

${\Omega }^{2}\left(U\right)←{𝔤}^{*}\left[1\right]:{F}_{A}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left\{{t}^{a}\right\}$ is a dual basis for ${𝔤}^{*}$ and $\left\{{r}^{a}\right\}$ the corresponding dual basis for ${𝔤}^{*}\left[1\right]$ and $\left\{{C}^{a}{}_{bc}\right\}$ the structure constants of the Lie bracket $\left[-,-\right]$ on $𝔤$, then the differential ${d}_{W\left(𝔤\right)}$ on the Weil algebra is defined on generators by

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

and

${d}_{W\left(𝔤\right)}{r}^{a}={C}^{a}{}_{bc}{t}^{b}\wedge {r}^{c}\phantom{\rule{thinmathspace}{0ex}}.$d_{W(\mathfrak{g})} r^a = C^a{}_{b c} t^b \wedge r^c \,.

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

${r}^{a}↦\left({F}_{A}{\right)}^{a}:=d{A}^{a}+\frac{1}{2}{C}^{a}{}_{bc}{A}^{b}\wedge {A}^{c}\phantom{\rule{thinmathspace}{0ex}}.$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

$d\left({F}_{A}{\right)}^{a}+{C}^{a}{}_{bc}{A}^{a}\wedge \left({F}_{A}{\right)}^{c}=0\phantom{\rule{thinmathspace}{0ex}}.$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 $𝔤$ be an arbitrary ∞-Lie-algebra and $W\left(𝔤\right)$ its Weil algebra. Then a collection of ∞-Lie algebra valued differential forms is a dg-algebra morphism

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

It curvature is the composite of morphism of graded vector space

${\Omega }^{•}\left(U\right)\stackrel{A}{←}W\left(𝔤\right)\stackrel{{F}_{\left(-\right)}}{←}{𝔤}^{*}\left[1\right]:{F}_{A}\phantom{\rule{thinmathspace}{0ex}}.$\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

${d}_{\mathrm{dR}}{F}_{A}+A\left({d}_{W\left(𝔤\right)}\left(-\right)\right)=0\phantom{\rule{thinmathspace}{0ex}}.$d_{dR} F_A + A(d_{W(\mathfrak{g})}(-)) = 0 \,.

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

## References

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

Revised on January 31, 2011 17:26:50 by Urs Schreiber (89.204.137.75)