nLab Chern-Simons form

Idea

A Chern-Simons form $CS(A)$ is a differential form naturally associated to a differential form $A \in \Omega^1(P,\mathfrak{g})$ with values in a Lie algebra $\mathfrak{g}$: it is the form trivializing (locally) a curvature characteristic form $\langle F_A \wedge \cdots \wedge F_A \rangle$ of $A$, for $\langle \cdots \rangle$ an invariant polynomial:

$$d_{dR} CS(A) = \langle F_A \wedge \cdots \wedge F_A \rangle \,,$$

where $F_A \in \Omega^2(X,\mathfrak{g})$ is the curvature 2-form of $A$.

Therefore it is often also called a secondary characteristic form.

More generally, for $A,A' \in \Omega^1(P, \mathfrak{g})$ two $\mathfrak{g}$-valued 1-forms and for $\hat A \in \Omega^1(P \times [0,1],\mathfrak{g})$ a “path of connections”, the Chern-Simons form relative to $A$ and $A'$ is a form that trivializes the difference between the two curvature characteristic forms

$$d_{dR}CS(A,A') = \langle (F_A)^k \rangle - \langle (F_{A'})^k \rangle \,.$$

Chern-Simons forms are of interest notably when the differential forms $A,A'$ are (local representatives of) connections on a $G$-principal bundle $P \to X$, for instance if $A \in \Omega^1(P,\mathfrak{g})$ is an Ehresmann connection 1-form.

Often the term Chern-Simons form is taken to refer to the case where $\mathfrak{g}$ is a semisimple Lie algebra with binary invariant polynomial $\langle -, -\rangle$ (e.g. the Killing form) in which case $CS(A)$ is the 3-form

$$\langle A \wedge d_{dR} A\rangle + c \langle A \wedge [A \wedge A] \rangle \,.$$

Even more specifically, often the term is understood to refer to the case where $\mathfrak{g} \subset \mathfrak{gl}(n)$ is a matrix Lie algebra, for instance $\mathfrak{o}(n)$ (for the orthogonal group) or notably $\mathfrak{u}(n)$ (for the unitary group). In that case the invariant polynomials may be taken to be given by matrix traces: $\langle \cdots \rangle = tr(\cdots )$.

Details

It is sufficient to discuss properties of Chern-Simons forms for $\mathfrak{g}$-valued 1-forms. The corresponding statements for connections on a $G$-bundle follow straightforwardly.

Paths of connections

Let $U$ be a smooth manifold.

Definition

A smooth path of $\mathfrak{g}$-valued 1-forms on $U$ is a smooth 1-form $\hat A \in \Omega^1(U\times [0,1],\mathfrak{g})$

Call this path pure shift if $\iota_{\partial_t} \hat A = 0$, where $t : U \times [0,1] \to [0,1] \hookrightarrow \mathbb{R}$ is the canonical coordinate along the interval.

We say this path goes from $A_0 := \psi_0^* \hat A$ to $A_1 := \psi_1^* \hat A$, where

$$\psi_t : U \simeq U \times * \stackrel{Id \times t}{\to} U \times [0,1]$$

picks the copy of $U$ at parameter $t$.

So a smooth path is a smooth 1-form on the cylinder $U \times [0,1]$ and it is pure shift if it has no “leg” along the $[0,1]$-direction. We will see that $\iota_{\partial_t} \hat A$ encodes infinitesimal gauge transformations, while $\partial_t \hat A$ is the change by infinitesimal shifts minus infinitesimal gauge transformations of the connection.

Definition/Observation

Let $P$ be an invariant polynomial on $\mathfrak{g}$ of arity $n$.

Consider the fiber integration

$$CS_P(A_0,A_1) := \int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \,.$$

This defines a $(2n-1)$-form $CS_P(A_0,A_1) \in \Omega^{2n-1}(U)$.

We have that the exterior differential of this form is the difference of the curvature characteristic forms of $A_0$ and $A_1$:

$$d_{dR} CS_P(A_0,A_1) = P(F_{A_1} \wedge \cdots \wedge F_{A_1}) - P(F_{A_0} \wedge \cdots \wedge F_{A_0})$$

Proof

Write the fiber integration more explicitly as an integral

$$CS_P(A_0,A_1) = \int_{[0,1]} \psi_t^* \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \,.$$

Then use that $d_{dR}$ is linear and commutes with pullback, use Cartan's magic formula $d_{dR} \circ \iota_{\partial_t} + \iota_{\partial} \circ d_{dR} = \mathcal{L}_{\partial_t}$ in view of the fact that $P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})$ is a closed form and then finally apply the Stokes theorem:

$$\begin{aligned} d_{dR} \int_0^1 \psi^*_t \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t & = \int_0^1 d_{dR} \psi^*_t \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \psi^*_t d_{dR} \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \psi^*_t \frac{d}{d t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \left(\frac{d}{d t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})\right)(t) d t \\ & = \psi^*_1 P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) - \psi^*_0 P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \\ &= P(F_{A_1} \wedge \cdots \wedge F_{A_1}) - P(F_{A_0} \wedge \cdots \wedge F_{A_0}) \end{aligned} \,.$$

Explicit formulas

Above we saw that a general expression for the Chern-Simons $CS_P(A_0,A_1)$ obtained from a path of connections $\hat A$ between $A_0$ and $A_1$ is

$$CS_P(A_0, A_1) = \int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \,.$$

We now unwind this to get explicit formulas for the Chern-Simons form in terms of wedge products of connection forms and their curvatures.

For $\hat A$ a pure shift path, $\hat A : t \mapsto A_t$ notice that the curvature 2-form of $\hat A$ is

$$F_{\hat A}(t) = F_{A_t} + (\partial_t A_t) \wedge d t \,.$$

Inserting this into the above expression yields

$$CS_P(A_0,A_1) = \int_0^1 P(\partial_t A \wedge F_{A_t} \wedge \cdots \wedge F_{A_t}) \,.$$

Notably if $A_0 = 0$ and $\hat A$ is the constant path $\hat A : t \mapsto t A$ to $A_1 := A$ such that

$$\begin{aligned} F_{\hat A} &= t d_{dR} A + t^2 [A \wedge A] \\ &= t F_A + (t^2 - t) [A \wedge A] \end{aligned}$$

this yields

$$CS_P(A) := \int_0^1 P(A \wedge (t F_{A} + (t^2 - t) [A \wedge A])) \wedge \cdots (t F_{A} + (t^2 - t) [A \wedge A]))) \,.$$

This is just an integral over a polynomial in $t$ with constant coefficients in forms. Peforming the integral yields a bunch of coefficients $c_i$ and with these the Chern-Simons form achieves the form

$$CS(A) = c_1 \langle A \wedge F_A \wedge \cdots \wedge F_A \rangle + c_2 \langle A \wedge [A \wedge A] \wedge F_A \wedge \cdots F_A \rangle + \cdots \,.$$

Particularly for $n = 2$ and using the definition of the curvature 2-form $F_A = d_{dR} A + [A \wedge A]$ we get

$$CS(A) = \langle A \wedge d A\rangle + c \langle A \wedge [A \wedge A]\rangle \,.$$

Gauged paths of connections

Above we defined $CS(A_0,A_1)$ for every path of connections form $A_0$ to $A_1$ which is pure shift . This is a possibly convenient but unnecessary restriction:

Notice that a general (gauged) path is a general 1-form $\hat A \in \Omega^1(U \times [0,1], \mathfrak{g})$ which we can decompose in the form

$$\hat A : t \mapsto A_t + \lambda d t \,,$$

where $\lambda$ is a $\mathfrak{g}$-valued function. The parallel transport of $\lambda d t$ along $[0,1]$ defines an element in $G$ and shift $(\partial_t A)_t$ of the connection along $[0,1]$ is now relative to the gauge transformation on $A$ induced by this function: the curvature 2-form now is

$$F_{\hat A} : t \mapsto F_{A_t} + ((\partial_t A)_t + d_U \lambda(t) + [\lambda,A_t]) \wedge d t$$

Proposition

The Chern-Simons form $\int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})$ defined with respect to any gauged lift of a pure shift path of connections differs from that of the pure shift path by an exact term.

In $\infty$-Chern-Weil theory

We discuss now a more encompassing perspective on Chern-Simons forms the way it occurs in ∞-Chern-Weil theory.

Prerequisites

We need to collect a few notions described elsewhere, on which the following discussion is based.

For $\mathfrak{g}$ a Lie algebra or more generally an ∞-Lie algebra we have the following dg-algebras naturally associated with it:

• the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$;

• the Weil algebra $W(\mathfrak{g})$;

• the algebra of invariant polynomials $inv(\mathfrak{g})$.

Given $n \in \mathbb{N}$, the Lie integration of $\mathfrak{g}$ to degree $n$ is the ∞-Lie groupoid which is the $n$-truncation of the simplicial presheaf

$$\exp(\mathfrak{g}) : U,[n] \mapsto dgAlg( CE(\mathfrak{g}), C^\infty(U)\otimes \Omega^\bullet(\Delta^n) ) \,,$$

where here and in the following $\Omega^\bullet(\Delta^n)$ denotes the de Rham complex dg-algebra of those smooth differential forms $\omega$ on the standard smooth $n$-simplex that have sitting instants in that for each $k \in \mathbb{N}$ every $k$-face of $\Delta^n$ has an open neighbourhood such that restricted to that neighbourhood $\omega$ is constant in the direction perpendicular to the face.

This is a one-object ∞-Lie groupoid which we may write

$$\mathbf{B}G = \mathbf{cosk}_{n+1} \exp(\mathfrak{g}) \,,$$

thus defining the ∞-Lie group $G$ that integrates $\mathfrak{g}$ in degree $n$.

At ∞-Chern-Weil theory is explained that a resolution of $\mathbf{B}G$ that serves to compute curvature characteristic forms in that it encodes pseudo-connections on $G$-principal ∞-bundles is given by the simplicial presheaf

$$\mathbf{B}G_{diff} := \mathbf{cosk}_{n+1} \left( U,[n] \mapsto \left\{ \array{ C^\infty(U) \otimes \Omega^\bullet(\Delta^n) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(\mathfrak{g}) } \right\} \right) \,,$$

where the vertical morphisms are the canonical ones.

Much of the subtlety of the full theory of connections of $\infty$-bundles comes from the finite coskeleton-truncation here. For the following discussion of Chern-Simons forms it is helpful to first ignore this issue by taking $n = \infty$, hence ignoring the truncation for the moment. This is sufficient for understand everything about Chern-Simons forms locally.

A cocycle in ∞-Lie algebra cohomology in degree $k$ is a morphism

$$CE(\mathfrak{g}) \leftarrow CE(b^{k-1} \mathbb{R}) : \mu \,.$$

Simply by composition (since we ignore the truncation for the moment), this integrates to a cocycle of the corresponding $\infty$-Lie groupoids

$$\exp(\mu) : \exp(\mathfrak{g}) \to \exp(b^{k-1}\mathbb{R}) \,,$$

At ∞-Chern-Weil theory it is discussed how the proper lift of this through the extension $\mathbf{B}G_{diff}$ that computes the abstractly defined curvature characteristic classes is given by finding the invariant polynomial $\langle -,-\rangle \in W(\mathfrak{g})$ that is in transgression with $\mu$ in that we have a commuting diagram

$$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs, \langle-,-\rangle))}{\leftarrow}& W(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle -,-\rangle}{\leftarrow}& CE(b^k \mathbb{R}) }$$

with a choice of interpolating Chern-Simons element $cs \in W(\mathfrak{g})$, which induces by precomposition with its upper part the morphism

$$\exp((cs,\langle-,-\rangle)) : \mathbf{B}G_{diff} \to \exp(b^{k-1}\mathbb{R})_{diff} \,.$$

By further projection to its lower part we get furthermore a morphism

$$\exp(b^{k-1}\mathbb{R})_{diff} \to \mathbf{\flat}_{dR}\mathbf{B}^k \mathbb{R}_{simp} := (U,[n] \mapsto \{ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow W(b^k \mathbb{R}) \}) \,.$$

Finally – and this is crucial now for obtaining the incarnation of Chern-Simons forms at integrals of curvature forms as in the above discussion – at ∞-Lie groupoid in the section on simplicial differential forms (see also circle n-bundles with connection the section Models from ∞-Lie integration) it is discussed that the operation that takes the $n$-cells on the right and integrates the corresponding forms over the $n$-simplex yields an equivalence

$$\int_{\Delta^\bullet} : (U,[n] \mapsto \{ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow W(b^k \mathbb{R}) \}) \;\;\;\;\to \;\;\;\; \Xi( \stackrel{d_{dR}}{\to}\Omega^{k-1}(-)\stackrel{d_{dR}}{\to}\Omega^k_{closed}(-))$$

to the image of the $\mathbb{R}$-Deligne complex of sheaves under the Dold-Kan correspondence.

Higher order Chern-Simons form

With all of the above in hand, we can make now the following observations:

For $X$ a smooth manifold and $\mathfrak{g}$ an ∞-Lie algebra with coefficient for pseudo-connections being $\mathbf{B}G_{diff}$ as above, a morphism

$$A : X \to \mathbf{B}G_{diff}$$

of simplicial presheaves (no resolution on the left, since we are concentrating on globally defined forms for the present purpose) is effectively a $\mathfrak{g}$-values differential form on $X$

For $\mu$ a cocycle on $\mathfrak{g}$ and $\langle -,-\rangle$ a corresponding invariant polynomial the composite

$$X \to \mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}$$

discussed above produces the corresponding curvature characteristic form.

A homotopy

$$(\nabla \to \nabla') : X \cdot \Delta[1] \to \mathbf{B}G_{diff}$$

is a smooth path in the space of $\mathfrak{g}$-valued forms on $X$. Under the adjunction

$$[X \cdot \Delta[1], \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}] \simeq [X, \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}^{\Delta[1]}]$$

this corresponds to a $(k-1)$-form on $X$ this is the Chern-Simons form

$$CS(\nabla \to \nabla') : X \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}^{\Delta[1]} \,.$$

The higher homotopies are higher order Chern-Simons forms.

The following proposition says this in a more precise way for ordinary Chern-Simons forms.

Ordinary Chern-Simons forms revisited

We now show how the traditional definition of Chern-Simons forms is reproduced by the general abstract mechanism.

Proposition

(ordinary Chern-Simons form)

Let $\mathfrak{g}$ be a Lie algebra, and $\langle -,-\rangle \in W(\mathfrak{g})$ an invariant polynomial.

Then morphisms (of simplicial presheaves)

$$A : X \to \mathbf{B}G_{diff}$$

are in canonical bijection with Lie-algebra valued 1-forms $A \in \Omega^1(X,\mathfrak{g})$. Morphisms

$$X \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}$$

are in canonical bijection with closed $k$-forms on $X$ and composition with the morphism

$$\mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}$$

discussed above and under this canonical identification the composite

$$\langle F_A \rangle : X \stackrel{A}{\to} \mathbf{B}G_{diff} \stackrel{}{\to} \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}$$

is the corresponding curvature characteristic form.

Homotopies

$$(A_1 \stackrel{\gamma}{\to} A_2) : X\times \Delta[1] \to \mathbf{B}G_{diff}$$

are in canonical bijection with smooth paths in the space of $\mathfrak{g}$-valued 1-forms on $X$ and under composition with $\mathbf{B}G_{diff} \to \mathbf{\flat}_{dR}\mathbf{B}^k \mathbb{R}$ these identify with the corresponding Chern-Simons form

$$\langle F_A\rangle \stackrel{CS(A \stackrel{\gamma}{\to} A')}{\to} \langle F_{A'}\rangle : X \to (\mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn})^{\Delta[1]} \,.$$

Proof

This is a straightforward unwinding of the definitions. We spell it out in the following in order to highlight the way the mechanism works.

By the Yoneda lemma and the definition of $\mathbf{B}G_{diff}$, a morphism $X \to \mathbf{B}G_{diff}$ is equivalently a diagram

$$\array{ C^\infty(X) \otimes \Omega^\bullet(\Delta^0) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) \otimes \Omega^\bullet(\Delta^0) &\stackrel{(A,F_A)}{\leftarrow}& W(\mathfrak{g}) } \,.$$

Since $CE(\mathfrak{g})$ is trivial in degree 0 and since $C^\infty(X)\otimess \Omega^\bullet(\Delta^0)$ is trivial above degree 0, the top morphism is necessarily 0 and the commutativity of the diagram is an empty condition.

The bottom morphism on the other hand enccodes precisely a $\mathfrak{g}$-valued form, as discussed in some detail at Weil algebra.

Composition with the morphism $\mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}$ is composition of the bottom morphism of the above digram with $W(\mathfrak{g}) \leftarrow CE(b^{k-1}\mathbb{R}) : \langle - \rangle$ followed by fiber integration of the resulting $k$-form

$$\Omega^\bullet(X)\otimes \Omega^\bullet(\Delta^0) \stackrel{(A,F_A)}{\leftarrow} W(\mathfrak{g}) \stackrel{\langle - \rangle}{\leftarrow} CE(b^{k-1}\mathbb{R}) : \langle F_A \rangle$$

over the point. This fiber integration is of course trivial, so that we find that indeed $X \stackrel{(A,F_A)}{\to} \mathbf{B}G_{diff} \stackrel{\langlw - \rangle}{\to} \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}$ is the curvature characteristic form defined on $\langle -\rangle$ on $A$.

Next, a homotopy $(A \stackrel{\gamma}{\to} A') : X \cdot \Delta[1] \to \mathbf{B}G_{diff}$ is (again by the Yoneda lemma) a diagram

$$\array{ C^\infty(X) \otimes \Omega^\bullet(\Delta^1) &\stackrel{\lambda}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) \otimes \Omega^\bullet(\Delta^0) &\stackrel{(A,F_A) \stackrel{\gamma}{\to} (A',F_{A'})}{\leftarrow}& W(\mathfrak{g}) } \,.$$

The top morphism defines an $X$-parameterized family of $\mathfrak{g}$-valued 1-form on the interval $[0,1]$, which is canonically identified with a smooth function $g : X \times [0,1] \to G$ into the simply connected Lie group integrating] $\mathfrak{g}$ based at the identity, $g(x,0) = e$, by the formula

$$\lambda = g^* \theta$$

where $\theta \in \Omega^1(G, \mathfrak{g})$ is the Maurer-Cartan form on $G$,

or conversely by parallel transport

$$f(x,s) = P \exp(\int_{[0,s]} \lambda(x,s) d s)$$

We may think of this as a smooth path of gauge transformations .

The bottom morphism encodes a $\mathfrak{g}$-valued form

$$\hat A + \lambda \in \Omega^1(X \times [0,1] , \mathfrak{g})$$

with $\hat A \in \Omega^1(X,\mathfrak{g}) \otimes C^\infty([0,1])$ and $\lambda$ as before, such that $\hat A(s = 0) = A$ and $\hat A(s = 1) = A'$.

This is a smooth path in the space of 1-forms . In the case that $\lambda = 0$ this is a pure shift path in the terminology above. we look at this case in the following, for ease of notation.

Under composition with $W(\mathfrak{g}) \leftarrow CE(b^{k-1}\mathbb{R}) : \langle -\rangle$ this becomes a $k$-form

$$\langle F_{\hat A } \rangle \in \Omega^{k}(X)\otimes C^\infty(\Delta^1)\oplus \Omega^{k-1}(X)\otimes \Omega^1(\Delta^1) \,.$$

The fiber integration of this over $\Delta^1$ is manifestly the same operation as that in the definition of the Chern-Simons form above.

As secondary characteristic forms

If a curvature characteristic form vanishes (for instance if the connection is flat or the degree of the curvature characteristic form is simply greater than the dimension of $X$) the corresponding Chern-Simons form is a closed form. So in this case the de Rham cohomology class of the curvature characteristic form becomes trivial, but the Chern-Simons form provides another de Rham class. This is therefore called a secondary characteristic class.

Chern-Simons theory

In particular on a 3-dimensional smooth manifold $X$ necessarily the Chern-Simons 3-form is closed. The functional

$$(A \in \Omega^1(X,\mathfrak{g})) \mapsto \int_X CS(A)$$

is the action functional of the quantum field theory called Chern-Simons theory.

More generally, for $X$ a $(2n-1)$-dimensional smooth manifold and $\langle -,\cdots, -\rangle$ an invariant polynomial of arity $n$, the analous formula defines the action functional of $(2n+1)$-dimensional Chern-Simons theory.

In terms of $\infty$-Lie algebroids

As discussed at invariant polynomial, Chern-Simons elements int the Weil algebra $W(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ induce the transgression between invariant polynomials and cocycles in Lie algebra cohomology.

For

$$\array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} \\ \downarrow && \downarrow \\ T P &\stackrel{(A,F_A)}{\to}& inn(\mathfrak{g}) \\ \downarrow && \downarrow \\ T X &\stackrel{(P_i(F_A))}{\to}& \prod_i b^{n_i} \mathfrak{u}(1) }$$

the data of an Ehresmann connection on a $G$-principal bundle expressed as a diagram of ∞-Lie algebroids with the curvature characteristic forms on the bottom, a choice of transgression element $cs_P$ for an invariant polynomial $P$ in transgression with a Lie algebra cocycle $\mu$ induces a diagram

$$\array{ \mathfrak{g} &\stackrel{\mu}{\to}& b^n \mathfrak{u}(1) \\ \downarrow && \downarrow \\ inn(\mathfrak{g}) &\stackrel{(cs_P,P)}{\to}& e b^{n} \mathfrak{u}(1) \\ \downarrow && \downarrow \\ \prod_i b^{n_i}\mathfrak{u}(1) &\stackrel{p_i}{\to}& b^{n+1} \mathfrak{u}(1) } \,.$$

The pasting of this to the above Ehresmann connection expresses in the middle horizontal morphism the Chern-Simons form $cs_P(A)$ and its curvature characteristic form $P(F_A)$

$$\array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} &\stackrel{\mu}{\to}& b^n \mathfrak{u}(1) \\ \downarrow && \downarrow && \downarrow \\ T P &\stackrel{A}{\to}& inn(\mathfrak{g}) &\stackrel{(cs_P,P)}{\to}& e b^n \mathfrak{u}(1) \\ \downarrow && \downarrow && \downarrow \\ T X &\stackrel{(P_i)}{\to}& \prod_i b^{n_i} \mathfrak{u}(1) &\stackrel{p_i}{\to}& b^{n+1} \mathfrak{u}(1) } \,.$$

Related concepts

• ∞-Lie algebroid valued differential forms

• connection on an ∞-bundle

• curvature

• Bianchi identity

• curvature characteristic form

• Chern-Simons form

References

The article introducing the concept is

• Shiing-Shen Chern, James Simons, Characteristic forms and geometric invariants The Annals of Mathematics, Second Series, Vol. 99, No. 1 (Jan., 1974), pp. 48-69 (jstor:1971013) (pdf)

As it says in the introduction of this article, it was motivated by an attempt to find a combinatorial formula for the Pontrjagin class of a Riemannian manifold (i.e. that associated to the O(n)-principal bundle to which the tangent bundle is associated) and the Chern-Simons form appeared as a boundary term that obstructed to original attempt to derive the Pontrjagin class by integrating curvature classes simplex-by-simplex. But A combinatorial formula of the kind these authors were looking for was however (nevertheless) given later in

• Jean-Luc Brylinski, Dennis McLaughlin? Čech cocycles for characteristic classes , Comm. Math. Phys. 178 (1996) (pdf)

The statements about “pure shift” paths are reviewed on the first few pages of

• James Simons, Dennis Sullivan, Structured vector bundles define differential K-theory (arXiv)

which discusses the relevance of Chern-Simons forms in differential K-theory.

The L-∞-algebra-formulation is discussed in SSS08.

An abstract algebraic model of the algebra of Chern’s characteristic classes and Chern-Simons secondary characteristic classes and of the gauge group action on this algebra (which also enables some noncommutative generalizations) is pioneered in 2 articles

• Israel M. Gelfand, Mikhail M. Smirnov, The algebra of Chern-Simons classes, the Poisson bracket on it, and the action of the gauge group, Lie theory and geometry, 261–288, Progr. Math. 123, Birkhäuser 1994; Chern-Simons classes and cocycles on the Lie algebra of the gauge group, The Gelfand Mathematical Seminars, 1993–1995, 101–122, Birkhäuser 1996.

On the geometric interpretation of the Chern-Simons invariant for more general principal bundles:

• Andreas Čap, Keegan J. Flood, Thomas Mettler: Flat extensions and the Chern-Simons 3-form (2024) [arXiv:2409.12811]