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:
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
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
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: $\langle \cdots \rangle = tr(\cdots )$.
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.
Let $U$ be a smooth manifold.
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
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.
Let $P$ be an invariant polynomial on $\mathfrak{g}$ of arity $n$.
Consider the fiber integration
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 $A_1$:
Write the fiber integration more explicitly as an integral
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:
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
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
Inserting this into the above expression yields
Notably if $A_0 = 0$ and $\hat A$ is the constant path $\hat A : t \mapsto t A$ to $A_1 := A$ such that
this yields
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
Particularly for $n = 2$ and using the definition of the curvature 2-form $F_A = d_{dR} A + [A \wedge A]$ we get
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
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
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.
We discuss now a more encompassing perspective on Chern-Simons forms the way it occurs in ∞-Chern-Weil theory.
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
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
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
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
Simply by composition (since we ignore the truncation for the moment), this integrates to a cocycle of the corresponding $\infty$-Lie groupoids
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
with a choice of interpolating Chern-Simons element $cs \in W(\mathfrak{g})$, which induces by precomposition with its upper part the morphism
By further projection to its lower part we get furthermore a morphism
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
to the image of the $\mathbb{R}$-Deligne complex of sheaves under the Dold-Kan correspondence.
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
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
discussed above produces the corresponding curvature characteristic form.
A homotopy
is a smooth path in the space of $\mathfrak{g}$-valued forms on $X$. Under the adjunction
this corresponds to a $(k-1)$-form on $X$ this is the Chern-Simons form
The higher homotopies are higher order Chern-Simons forms.
The following proposition says this in a more precise way for ordinary Chern-Simons forms.
We now show how the traditional definition of Chern-Simons forms is reproduced by the general abstract mechanism.
(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)
are in canonical bijection with Lie-algebra valued 1-forms $A \in \Omega^1(X,\mathfrak{g})$. Morphisms
are in canonical bijection with closed $k$-forms on $X$ and composition with the morphism
discussed above and under this canonical identification the composite
is the corresponding curvature characteristic form.
Homotopies
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
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
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
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
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
where $\theta \in \Omega^1(G, \mathfrak{g})$ is the Maurer-Cartan form on $G$,
or conversely by parallel transport
We may think of this as a smooth path of gauge transformations .
The bottom morphism encodes a $\mathfrak{g}$-valued form
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
The fiber integration of this over $\Delta^1$ is manifestly the same operation as that in the definition of the Chern-Simons form above.
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.
In particular on a 3-dimensional smooth manifold $X$ necessarily the Chern-Simons 3-form is closed. The functional
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.
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
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
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 $P(F_A)$
Chern-Simons form
The article introducing the concept is
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
The statements about “pure shift” paths are reviewed on the first few pages of
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