Schreiber
curvature of ∞-Lie algebroid valued differential forms

Context

1. differential cohomology in an (∞,1)-topos -- survey

2. structures in an (∞,1)-topos

3. path ∞-groupoid

4. Lie theory in an (∞,1)-topos

5. differential cohomology in an (∞,1)-topos

6. differential cohomology in an (∞,1)-topos -- examples

7. differential cohomology in an (∞,1)-topos -- references

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Idea

The curvature of ∞-Lie algebroid valued differential forms is a measure for their cohomological non-triviality. It generalizes the notion of curvature for Lie algebra value forms to that if differential forms with values in L-∞ algebras and, more generally ∞-Lie algebroids.

Definition

For the discussion we place ourselves in the context of a smooth (∞,1)-topos $H$ and use the ∞-Lie theory available there.

Direct description

For $𝔞$ an ∞-Lie algebroid, a morphism

$$\omega :{\Pi }_{\inf }(X)\to 𝔞$$\omega : \mathbf{\Pi}_{inf}(X) \to \mathfrak{a}

from the infinitesimal path ∞-groupoid of some $X$ into $𝔞$ is a collection of flat ∞-Lie algebroid valued differential forms.

With $T𝔞:={\Pi }_{\inf }(𝔞)$ the tangent Lie algebroid of $𝔞$ itself, we have that

$$\omega :{\Pi }_{\inf }(X)\to T𝔞$$\omega : \mathbf{\Pi}_{inf}(X) \to T \mathfrak{a}

is a collection of not-necessarily flat $𝔞$-valued differential forms.

…

This collection of $\Sigma 𝔞$-valued differential forms $P(F_{\omega })$ is the curvature characteristic forms of $\omega$.

The curvature $F_{\omega }$ itself here denotes the “coned part” of $\omega$. This is extracted cleanly by passing from ∞-Lie algebroids to their Chevalley-Eilenberg algebras

dg-Algebra description

The sequence of Chevalley-Eilenberg dg-algebras corresponding to $𝔞\to T𝔞\to \Sigma 𝔞$ is

$$\mathrm{CE}(𝔞)\leftarrow W(𝔞)\leftarrow \mathrm{inv}(𝔞)$$CE(\mathfrak{a}) \leftarrow W(\mathfrak{a}) \leftarrow inv(\mathfrak{a})

where

• $\mathrm{CE}(𝔞)$ is the plain Chevalley-Eilenberg algebra of $𝔞$;

• $W(𝔞)$ is the Weil algebra of $𝔞$;

• $\mathrm{inv}(𝔞)$ is the algebra of invariant polynomials on $𝔞$.

Curvature forms

Concretely we have with the semi-free dga

$$\mathrm{CE}(𝔞)=({\wedge }_{C^{\mathrm{\infty }}({𝔞}_0)}^{•}{V}^{*},d_{𝔞})$$CE(\mathfrak{a}) = (\wedge_{C^\infty(\mathfrak{a}_0)}^\bullet V^*, d_\mathfrak{a})

that

$$\mathrm{CE}(𝔞)=({\wedge }_{C^{\mathrm{\infty }}({𝔞}_0)}^{•}{V}^{*}\oplus V^{*}[-1],d_{T𝔞})$$CE(\mathfrak{a}) = ( \wedge_{C^\infty(\mathfrak{a}_0)}^\bullet V^* \oplus V^*[-1], d_{T \mathfrak{a}})

consists of the original and one shifted copy of the underlying graded vector space $V^{*}$.

Then for

$${\Omega }^{•}(X)\stackrel{\omega }{\leftarrow }W(𝔞)$$\Omega^\bullet(X) \stackrel{\omega}{\leftarrow} W(\mathfrak{a})

a collection of not-necessarily flat $𝔞$-valued differential forms, their curvature forms $F_{\omega }$ is the composite in the category of underlying algebras

$${\Omega }^{•}(X)\stackrel{\omega }{\leftarrow }W(𝔞)\leftarrow {\wedge }^{•}{V}^{*}[-1]:F_{\omega }.$$\Omega^\bullet(X) \stackrel{\omega}{\leftarrow} W(\mathfrak{a}) \leftarrow \wedge^\bullet V^*[-1] : F_\omega \,.

Notice that when $𝔤$ is an Lie n-algebra then $F_{\omega }$ has

• a 2-form curvature component $F_{\omega \mathrm{,2}}\in {\Omega }^2(X,V_1)$;

• a 3-form curvature component $F_{\omega \mathrm{,3}}\in {\Omega }^3(X,V_2)$;

• and so on, up to…

• an $(n+1)$-form curvature component $F_{\omega ,n+1}\in {\Omega }^{n+1}(X,V_n)$.

Remark

(“fake curvature”) Especially in the case $n=2$ in the context of principal 2-bundles with connection, the 2-form curvature has sometimes been called fake curvature . This terminology was motivated from the fact that for an ordinary connection on a bundle, only a 2-form curvature piece exists, as this corresponds to the case where $𝔤$ is a Lie 1-algebra. For general $𝔤$ there is no real sense in which the top-degree curvature form were more genuinely a curvature form than the lower degree forms. All of them together constitute the curvature form data of a non-flat -Lie ∞-Lie algebroid valued differential forms.

Bianchi identity

For $\nu \in V^{*}$ a homogeneous element with $V^{*}$ as above, we say that the commutativity of

$$\begin{array}{ccc}\nu & \stackrel{\omega }{\mapsto }& F_{\omega }(\nu )\\ {\downarrow }^{d_{𝔞}}& & {\downarrow }^{d_{\mathrm{dR}}}\\ d_{𝔞}\nu & \stackrel{\omega }{\mapsto }& d{F}_{\omega }(\nu )=\omega (d_{𝔞}\nu )\end{array}$$\array{ \nu &\stackrel{\omega}{\mapsto}& F_\omega(\nu) \\ \downarrow^{d_{\mathfrak{a}}} && \downarrow^{d_{dR}} \\ d_{\mathfrak{a}}\nu &\stackrel{\omega}{\mapsto}& d F_\omega(\nu) = \omega(d_{\mathfrak{a}} \nu) }

given by the fact that $\omega$ is a morphism of dg-algebras is the Bianchi identity satisfied by the curvature forms $F_{\omega }$.

Twisted Bianchi identity

Morphisms $\omega :{\Pi }^{\inf }(X)\to T𝔞$ encoding ∞-Lie algebroid valued differential forms arise as Cartan-Ehresmann ∞-connections on principal ∞-bundles. In applications – see for instance the discussion at differential twisted String and Fivebrane structures – these $\mathrm{\infty }$-bundles with connections are typically not plain differential cocycles, but twisted differential cocycles.

This means in particular that

1. $𝔞$ itself is part of a fibration sequence

$$𝔞\to 𝔟\to 𝔠$$\mathfrak{a} \to \mathfrak{b} \to \mathfrak{c}

of ∞-Lie algebroids

1. there is a prescribed twist given by $𝔠$-valued differential forms $\lambda :{\Pi }^{\inf }\to T𝔠$

2. a collection of $\lambda$-twisted $𝔞$-valued differential forms is a collection of $𝔟$-valued differential forms $\omega :{\Pi }^{\inf }(X)\to 𝔟$ such that the underlying $𝔠$-forms are given by $\lambda$, i.e. such that $\omega$ fits into the diagram

   $$\begin{array}{ccc}& & 𝔟\\ & {}^{\omega }\nearrow & & \searrow \\ {\Pi }^{\inf }(X)& & \stackrel{\lambda }{\to }& & 𝔠\end{array}$$\array{ && \mathfrak{b} \\ & {}^{\omega}\nearrow && \searrow \\ \Pi^{inf}(X) &&\stackrel{\lambda}{\to}&& \mathfrak{c} }

   or dually

   $$\begin{array}{ccc}& & \mathrm{CE}(𝔟)\\ & {}^{\omega }\swarrow & & \nwarrow \\ {\Omega }^{•}(X)& & \stackrel{\lambda }{\to }& & W(𝔠)\end{array}$$\array{ && CE(\mathfrak{b}) \\ & {}^{\omega}\swarrow && \nwarrow \\ \Omega^\bullet(X) &&\stackrel{\lambda}{\to}&& W(\mathfrak{c}) }

Then one says that the twisted Bianchi identity of $\omega$ regarded as $\lambda$-twisted $𝔞$-valued differential forms is the (ordinary) Bianchi identity of $\omega$ regarded as $𝔟$-valued differential forms.

Curvature characteristic forms

While the notion of curvature forms is a useful one in applications, it is strictly not an invariant notion as it is a bit evil.

The invariant intrinsic information on the curvature of ∞-Lie algebroid valued differential forms ${\Omega }^{•}(X)\leftarrow W(𝔞):\omega$ is instead measured by the curvature characteristic forms $P(F_{\omega })$ given by the composite

$${\Omega }^{•}(X)\stackrel{\omega }{\leftarrow }W(𝔞)\leftarrow \mathrm{inv}(𝔞):P(F_{\omega }).$$\Omega^\bullet(X) \stackrel{\omega}{\leftarrow} W(\mathfrak{a}) \leftarrow inv(\mathfrak{a}) : P(F_\omega) \,.

Examples

Lots of details on this are in

• Hisham Sati, U.S., Jim Stasheff, $L_{\mathrm{\infty }}$-connections; in Recent developments in QFT, Birkhäuser (2008), (arXiv)

Ordinary Lie 1-algebra valued forms

Here we spell out the example of 1-forms with values in an ordinary Lie algebra. And to facilitate comparison with (large) parts of the literature, we spell out everything in component formulas after choosing some basis.

So consider an ordinary Lie algebra $𝔤$ with Lie bracket $[-,-]:𝔤\times 𝔤\to 𝔤$.

We describe the familiar notion of $𝔤$-valued differential forms and their curvatures and Bianchi identities as a special case of the above general nonsense.

The Chevalley-Eilenberg algebras

Recall first of all that and how the Lie algebra $𝔤$ is equivalently encoded in its Chevalley-Eilenberg dg-algebra: this is the Grassmann algebra

$${\wedge }^{•}{𝔤}^{*}=ℝ\oplus {𝔤}^{*}\oplus ({𝔤}^{*}\wedge {𝔤}^{*})\oplus ({𝔤}^{*}\wedge {𝔤}^{*}\wedge {𝔤}^{*})\oplus \cdots$$\wedge^\bullet \mathfrak{g}^* = \mathbb{R} \oplus \mathfrak{g}^* \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^*) \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^*\wedge \mathfrak{g}^*) \oplus \cdots

on the dual vector space underlying the Lie algebra, equipped with the derivation $d_{𝔤}$ wich is simply the dual of the Lie bracket

$$d:{𝔤}^{*}\stackrel{[-,-{]}^{*}}{\to }{𝔤}^{*}\wedge {𝔤}^{*}$$d : \mathfrak{g}^* \stackrel{[-,-]^*}{\to} \mathfrak{g}^* \wedge \mathfrak{g}^*

extended by the graded Leibnitz rule to all of ${\wedge }^{•}{𝔤}^{*}$.

So if we choose a basis $\{t_a\}$ of $𝔤$ and a corresponding dual basis $\{t^a\}$ of ${𝔤}^{*}$ then with $\{C^a{}_{bc}\}$ the structure constants of the Lie bracket in this basis

$$[t_b,t_c]=C_{bc}^a{t}_a$$[t_b, t_c] = C^a_{b c} t_a

we have that ${\wedge }^{•}{𝔤}^{*}$ is just the algebra obtained by formally wedging the $t^a$ together in expressions like $5t^a\wedge t^b-2t^c$, such that we throw in a sign when two of them change place

$$t^a\wedge t^b=-t^b\wedge t^a$$t^a \wedge t^b = - t^b \wedge t^a

(we regard the $t^a$s as being of degree 1, like a 1-form) and on which the differential $d$ acts as

$$d{t}^a=-\frac{1}{2}{C}_{bc}^a{t}^b\wedge t^c.$$d t^a = -\frac{1}{2} C^a_{b c} t^b \wedge t^c \,.

Recall again that this is just the Lie bracket dualized and expressed in a basis. One advantage of this dualization is that the Jacobi identity becomes an easy statement this way: it is equivalent to the statement that the differential squares to 0, i.e. $d^2=0$.

One may want to check this explicitly in the chosen basis, using the above formulas:

$$\begin{array}{rl}dd{t}^a& =d(-\frac{1}{2}{C}_{bc}^a{t}^b\wedge t^c)\\ & =-\frac{1}{2}{C}_{bc}^a(-\frac{1}{2}{C}_{de}^b{t}^d\wedge t^e)\wedge t^c-\frac{1}{2}{C}_{bc}^a{t}^b\wedge (-\frac{1}{2}{C}_{de}^c{t}^d\wedge t^e)\\ & \propto C_{bc}^a{C}_{de}^b{t}^c\wedge t^d\wedge t^e\\ & =C_{b[c}^a{C}_{de]}^b{t}^c\wedge t^d\wedge t^e\end{array}$$\begin{aligned} d d t^a &= d (- \frac{1}{2} C^a_{b c} t^b \wedge t^c) \\ &= - \frac{1}{2}C^a_{b c} (-\frac{1}{2}C^b_{d e} t^d \wedge t^e)\wedge t^c - \frac{1}{2} C^a_{b c} t^b \wedge (-\frac{1}{2} C^c_{d e} t^d \wedge t^e) \\ & \propto C^a_{b c} C^b_{d e} t^c \wedge t^d \wedge t^e \\ & = C^a_{b [c} C^b_{d e]} t^c \wedge t^d \wedge t^e \end{aligned}

where we used the above definition, the graded Leibniz rule satisfied by $d$ (by definition) and then in the last line denoted by square brackets on the indices the antisymmetrization of these indices, as usual in the component notation. And indeed, the expression

$$C_{b[c}^a{C}_{de]}^b=0$$C^a_{b [c} C^b_{d e]} = 0

is the expression of the Jacobi identity of the Lie bracket in component form.

The Grassmann algebra ${\wedge }^{•}{𝔤}^{*}$ together with this differential form the dg-algebra that is called the Chevalley-Eilenberg algebra

$$\mathrm{CE}(𝔤)=({\wedge }^{•}{𝔤}^{*},d)$$CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^* , d)

of the Lie algebra $𝔤$.

Using just this dg-algebra we can already talk in a nice way about $𝔤$-valued differential forms.

$𝔤$-Valued forms

Consider some manifold $X$ and the deRham complex ${\Omega }^{•}(X)$. This is also a dg-algebra, if course.

But let us disregard the deRham differential and the above differential on $\mathrm{CE}(𝔤)$ for a moment and just regarded the underlying graded commutative algebras with their respective wedge products. Then consider morphisms of graded commutative algebras

$${\Omega }^{•}(X)\leftarrow \mathrm{CE}(𝔤):A.$$\Omega^\bullet(X) \leftarrow CE(\mathfrak{g}) : A \,.

Such a morphism is supposed respect the grading. So the generator $t^a$ of $\mathrm{CE}(𝔤)$, which is of degree 1, has to map to some degree 1 element in ${\Omega }^{•}(X)$, i.e. to a 1-form. This 1-form we can conveniently call $A^a$:

$$A:(t^a\in {\wedge }^{•}{𝔤}^{*})\mapsto (A^a\in {\Omega }^1(X)).$$A : (t^a \in \wedge^\bullet \mathfrak{g}^*) \mapsto (A^a \in \Omega^1(X)) \,.

And now, since a Grassmann algebra is a free graded commutative algebra, this choice already fixes the entire morphism $A:\mathrm{CE}(𝔤)\to {\Omega }^{•}(X)$ all wedge products of generators of the Chevalley-Eilenberg algebra have to map to the corresponding wedge products of 1-forms, for instance:

$$A:t^a\wedge t^b\mapsto A^a\wedge A^b,$$A : t^a \wedge t^b \mapsto A^a \wedge A^b \,,

where now on the right we have the wedge product of differential forms on $X$.

So we have have found that $A$ is, for each basis element $t_a$ of $𝔤$, given by a 1-form $A^a\in {\Omega }^1(X)$. We may collect these 1-forms together and write

$$A=A^a\otimes t_a\in {\Omega }^1(X)\otimes 𝔤.$$A = A^a \otimes t_a \in \Omega^1(X) \otimes \mathfrak{g} \,.

This is the $𝔤$-valued 1-form. The space

$${\Omega }^{•}(X,𝔤)={\Omega }^{•}(X)\otimes 𝔤$$\Omega^\bullet(X,\mathfrak{g}) = \Omega^\bullet(X) \otimes \mathfrak{g}

is the space of $𝔤$-valued 1-forms on $X$.

Curvature of $𝔤$-valued 1-forms

For the above derivation of the morphism $A$ as a $𝔤$-valued 1-form we haven’t yet used that the algebras in question here are equipped with differentials. We want to check to which degree the morphism $A$ respects these differentials. We say it does respect them if it doesn’t matter whether we first apply the morphism

$$t^a\stackrel{A}{\mapsto }{A}^a$$t^a \stackrel{A}{\mapsto} A^a

and then the deRham differential

$$A^a\stackrel{d_{\mathrm{dR}}}{\mapsto }{d}_{\mathrm{dR}}{A}^a$$A^a \stackrel{d_{dR}}{\mapsto} d_{dR} A^a

or first the Chevalley-Eilenberg differential

$$t^a\stackrel{d_{\mathrm{CE}(𝔤)}}{\mapsto }-\frac{1}{2}{C}_{bc}^a{t}^b\wedge t^c$$t^a \stackrel{d_{CE(\mathfrak{g})}}{\mapsto} -\frac{1}{2}C^a_{b c} t^b \wedge t^c

and then the morphism

$$-\frac{1}{2}{C}_{bc}^a{t}^b\wedge t^c\stackrel{A}{\mapsto }-\frac{1}{2}{C}_{bc}^a{A}^b\wedge A^c.$$-\frac{1}{2}C^a_{b c} t^b \wedge t^c \stackrel{A}{\mapsto} -\frac{1}{2}C^a_{b c} A^b \wedge A^c \,.

We say the differentials are respected if these two results coincide, i.e. if the 2-forms

$$F_A^a:=d_{\mathrm{dR}}{A}^a+\frac{1}{2}{C}_{bc}^a{A}^b\wedge A^c$$F_A^a := d_{dR} A^a + \frac{1}{2} C^a_{b c} A^b \wedge A^c

vanishes. These 2-forms are the components of the curvature 2-form $F_A$ of $A$. If they vanish, this means that the diagram

$$\begin{array}{ccc}{\Omega }^{•}(X)& \stackrel{A}{\leftarrow }& \mathrm{CE}(𝔤)\\ {\uparrow }^{d_{\mathrm{dR}}}& & {\uparrow }^{d_{\mathrm{CE}(𝔤)}}\\ {\Omega }^{•}(X)& \stackrel{A}{\leftarrow }& \mathrm{CE}(𝔤)\end{array}$$\array{ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow^{\rlap{d_{dR}}} && \uparrow^{\rlap{d_{CE(\mathfrak{g})}}} \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& CE(\mathfrak{g}) }

commutes.

Geometric interpretation of the curvature

The algebraic relations that we have derived as a very obvious geometric interpretation. from the discussion at Chevalley-Eilenberg algebra we know that all these algebras that we are dealing with are naturally thought of as being the algebra of funcions on the space of morphisms of certain ∞-Lie groupoids. For our simple special case here this is very simple:

The deRham complex may be thought of as the algebra of functions on the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)$. This has as objects the points of $X$, as morphisms infinitesimal paths

$$x\to y$$x \to y

in $X$

as 2-morphisms infinitesimal little surfaces

$$\begin{array}{ccc}x& \to & y_1\\ \downarrow & \searrow & \downarrow \\ y_2& \to & z\end{array}$$\array{ x &\to& y_1 \\ \downarrow & \searrow & \downarrow \\ y_2 &\to& z }

in $X$, and so on.

On the other hand, $\mathrm{CE}(𝔤)$ is the algebra of functions on the infinitesimal version $B{G}_{(1)}$ of what is called the delooping groupoid $BG$ of the Lie group of which $𝔤$ is a Lie algebra. This has a single object $*$, and a morphism is an infinitesimal group element

$$*\stackrel{e+{\lambda }^a{t}_a}{\to }*$${*} \stackrel{e + \lambda^a t_a}{\to} {*}

for $e$ the neutral element of the group (the identity), for $t_a$ an element of the Lie algebra as before and for ${\lambda }^a$ some coefficient.

A 2-morphism is an infinitesimal surface bounded by such infinitesimal 1-morphisms such that going either way around the surface

$$\begin{array}{ccc}*& \stackrel{e+{\lambda }_1^a{t}_a}{\to }& *\\ {\downarrow }^{{\lambda }_3^a{t}_a}& \searrow & {\downarrow }^{{\lambda }_2^a{t}_a}\\ *& \stackrel{{\lambda }_4^a{t}_a}{\to }& *\end{array}$$\array{ {*} &\stackrel{e + \lambda^a_1 t_a}{\to}& {*} \\ \downarrow^{\rlap{\lambda_3^a t_a}} & \searrow & \downarrow^{\rlap{\lambda_2^a t_a}} \\ {*} &\stackrel{\lambda_4^a t_a}{\to}& {*} }

produces the same result when then morphisms are composed using the product in the Lie group: the top right way around the square here yields

$$(e+{\lambda }_1^a{t}_a))(e+{\lambda }_2^b{t}_b))=e+{\lambda }_1^a{t}_a+{\lambda }_2^b{t}_b+{\lambda }_1^a{\lambda }_2^b{t}_a{t}_b$$(e + \lambda_1^a t_a) )(e + \lambda_2^b t_b) ) = e + \lambda_1^a t_a + \lambda_2^b t_b + \lambda_1^a \lambda_2^b t_a t_b

and the other way round yields

$$(e+{\lambda }_3^a{t}_a))(e+{\lambda }_4^b{t}_b))=e+{\lambda }_3^a{t}_a+{\lambda }_4^b{t}_b+{\lambda }_3^a{\lambda }_4^b{t}_a{t}_b.$$(e + \lambda_3^a t_a) )(e + \lambda_4^b t_b) ) = e + \lambda_3^a t_a + \lambda_4^b t_b + \lambda_3^a \lambda_4^b t_a t_b \,.

A morphism of dg-algebras of the form we have been considering

$${\Omega }^{•}(X)\leftarrow \mathrm{CE}(𝔤):A$$\Omega^\bullet(X) \leftarrow CE(\mathfrak{g}) : A

is now evidently equivalenty a morphism

$$A:{\Pi }^{\inf }(X)\to B{G}_{(1)}$$A : \Pi^{inf}(X) \to \mathbf{B}G_{(1)}

that sends infinitesimal paths in $X$ to infinitesimal group elements of the form $e+{\lambda }^a{t}_a$:

$$A:(x\to y)\mapsto (*\stackrel{e+A^a(x,y)t_a}{\to }*).$$A : (x \to y) \;\;\mapsto\;\; ({*} \stackrel{e + A^a(x,y) t_a}{\to} {*}) \,.

If we denote by

$$v=y-x$$v = y - x

the tangent vector that connects the infinitesimally close points $x$ and $y$ and write $A(x,y)=A_x(v)$ as a function of the first point and the vector pointing away from it, then this reads

$$A:(x\to y)\mapsto (*\stackrel{e+A_x^a(v)t_a}{\to }*).$$A : (x \to y) \;\;\mapsto\;\; ({*} \stackrel{e + A^a_x(v) t_a}{\to} {*}) \,.

We can now look at what this assignment $A$ of infinitesimal group elements to infinitesimal paths does to a little square in $X$ as above, with sides spanned by tangent vectors $v_1$ and $v_2$. We find

$$A:\begin{array}{ccc}x& \stackrel{v_1}{\to }& y_1\\ {\downarrow }^{v_2}& \searrow & \downarrow \\ y_2& \to & z\end{array}\mapsto \begin{array}{ccc}*& \stackrel{e+A_x(v_1)t_a}{\to }& *\\ {\downarrow }^{A_x(v_2{)}^a{t}_a}& \searrow & {\downarrow }^{A_{y_1}(v_2{)}^a{t}_a}\\ *& \stackrel{A_{y_2}(v_1{)}^a{t}_a}{\to }& *\end{array}.$$A \;\; : \;\; \array{ x &\stackrel{v_1}{\to}& y_1 \\ \downarrow^{v_2} & \searrow & \downarrow \\ y_2 &\to & z } \;\;\mapsto\;\; \array{ {*} &\stackrel{e + A_x(v_1) t_a}{\to}& {*} \\ \downarrow^{\rlap{A_x(v_2)^a t_a}} & \searrow & \downarrow^{\rlap{A_{y_1}(v_2)^a t_a}} \\ {*} &\stackrel{A_{y_2}(v_1)^a t_a}{\to}& {*} } \,.

For the result on the right to qualify as a 2-morphism in $B{G}_{(1)}$ we need that

going around the top right edges, which yields

$$e+A_x^a(v_1)t_a+A_{y_1}^b(v_2)t_b+A_x^a(v_1)A_{y_1}^b(v_2)t_a{t}_b$$e + A_x^a(v_1) t_a + A^b_{y_1}(v_2) t_b + A_x^a(v_1) A^b_{y_1}(v_2) t_a t_b

is the same as

$$e+A_x^a(v_2)t_a+A_{y_2}^b(v_1)t_b+A_x^a(v_2)A_{y_2}^b(v_1)t_a{t}_b.$$e + A_x^a(v_2) t_a + A^b_{y_2}(v_1) t_b + A_{x}^a(v_2) A^b_{y_2}(v_1) t_a t_b \,.

To express what this means as a condition at the point $x$, we may Taylor expand to first order

$$A_{y_1}=A_x+{\partial }_{v_1}{A}_x$$A_{y_1} = A_x + \partial_{v_1} A_x

and

$$A_{y_2}=A_x+{\partial }_{v_2}{A}_x.$$A_{y_2} = A_x + \partial_{v_2} A_x \,.

Then some terms cancel and the above condition becomes, to second order

$${\partial }_{v_1}{A}_x^a(v_2)t_a+A_x^a(v_1)A_x^b(v_2)t_a{t}_b={\partial }_{v_2}{A}_x^a(v_1)t_a+A_x^a(v_2)A_x^b(v_1)t_a{t}_b.$$\partial_{v_1} A^a_x(v_2) t_a + A_x^a(v_1) A^b_{x}(v_2) t_a t_b = \partial_{v_2} A^a_x(v_1) t_a + A_x^a(v_2) A^b_{x}(v_1) t_a t_b \,.

In other words, the expression

$$F_A(v_1,v_2):={\partial }_{v_1}{A}_x^a(v_2)t_a-{\partial }_{v_2}{A}_x^a(v_1)t_a+\frac{1}{2}{A}_x^a(v_1)A_x^b(v_2)[t_a,t_b]$$F_A(v_1,v_2) := \partial_{v_1} A^a_x(v_2) t_a - \partial_{v_2} A^a_x(v_1) t_a + \frac{1}{2} A_x^a(v_1) A^b_{x}(v_2) [t_a, t_b]

has to vanish. This is the curvature form that we already found above by more algebraic means.

If this does not vanish, then we don’t really have a morphism $A:{\Pi }^{\inf }(X)\to B{G}_{(1)}$. But then we instead have some morphism that uses the 1-forms $A^a$ to assigns data to little edges, and that uses the 2-forms $F_A^a$ to assign data to little surfaces. That morphism then will respect a condition as above, but now on little cubes. That condition is the Bianchi identity

$$d{F}_A+[A\wedge F_A]=0$$d F_A + [A \wedge F_A] = 0

on the curvature 2-form.

(… so much for the moment, to be continued …)

Examples with twisted Bianchi identities

This section discussess examples of $\mathrm{\infty }$-Lie algebroid valued forms with twisted Bianchi identities. Some of the twisted examples appearing in physics are discussed at differential twisted String and Fivebrane structures.

Simplest abelian example

Let $𝔞=b^n𝔲(1)=\mathrm{Lie}(B^{n}U(1))$ be the L-∞ algebra of the $n$-fold delooping of the Lie group $U(1)\simeq ℝ/\mathbb{Z}$.

Ordinary case

Then $\mathrm{CE}(𝔞)$ is the Eilenberg-MacLane dg-algebra $K(n,ℝ)$ and a flat $b^n𝔲(1)$-valued differential form datum

$${\Omega }^{•}(X)\leftarrow \mathrm{CE}(b^n𝔲(1))$$\Omega^\bullet(X) \leftarrow CE(b^n \mathfrak{u}(1))

is precisely a closed n-form on $X$. A non-flat $b^n𝔲(1)$-valued differential form datum

$${\Omega }^{•}(X)\leftarrow W(b^n𝔲(1)):\omega$$\Omega^\bullet(X) \leftarrow W(b^n \mathfrak{u}(1)) : \omega

is precisely an arbitrary $n$-form on $X$. Its single non-vanishing curvature component is the $(n+1)$-form curvature

$$F_{\omega }(n+1)=d_{\mathrm{dR}}\omega .$$F_\omega(n+1) = d_{dR} \omega \,.

The Bianchi identity this satisfies is simply

$$d{F}_{\omega }(n+1)=0.$$d \; F_\omega(n+1) = 0 \,.

Twisted case

Consider twisting with respect to the fibration sequence

$$b^n𝔲(1)\to T{b}^n𝔲(1)\to b^{n+1}𝔲(1)$$b^n \mathfrak{u}(1) \to T b^n \mathfrak{u}(1) \to b^{n+1}\mathfrak{u}(1)

With respect to this a twisting form datum is an $(n+1)$-form $\lambda$. A $\lambda$-twisted $b^n𝔲(1)$-valued differential form datum is an $n$-form $\omega$ but with curvature forms

$$F_{\omega }(n+1)=d\omega +\lambda$$F_\omega(n+1) = d \omega + \lambda

$$F_{\omega }(n+2)=d\lambda .$$F_\omega(n+2) = d \lambda \,.

The Bianchi identity which in the untwisted case was $d{F}_{\omega }(n+1)=0$ is now the twisted Bianchi identity

$$d{F}_{\omega }(n+1)=F_{\omega }(n+2).$$d \; F_\omega(n+1) = F_\omega(n+2) \,.

Twisted $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$-valued forms

details to come, here a quick outline:

by the above, an ordinary $b𝔲(1)$-valued form is a 2-form $B$ with 3-form curvature $H=dB$. The ordinary Bianchi identity for this is $dH=0$.

A twisted incarnation of this appears as part of the datum of tisted $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$-forms. The other part of this is an $\mathrm{𝔰𝔬}(n)$-valued 1-form $\omega$. The twisted Bianchi identity in this case for the 3-form curvature is $dH=\frac{1}{2}{p}_1(F_{\omega })=⟨F_{\omega }\wedge F_{\omega }⟩$, where on the right we have the Pontryagin 4-form of $F_{\omega }=d\omega +[\omega \wedge \omega ]$.

this (in a slightly more refined form) is known part of the differential form structure of the Green-Schwarz mechanism.