nLab
Bianchi identity

Context

\infty-Chern-Weil theory

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(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.

Definition

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

and

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.

References

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

Revised on September 25, 2014 06:50:02 by Chenchang Zhu (134.76.82.123)