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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Chern-Simons form} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory} [[!include infinity-Chern-Weil theory - contents]] \hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology} [[!include differential cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{Details}{Details}\dotfill \pageref*{Details} \linebreak \noindent\hyperlink{PathsOfConnections}{Paths of connections}\dotfill \pageref*{PathsOfConnections} \linebreak \noindent\hyperlink{explicit_formulas}{Explicit formulas}\dotfill \pageref*{explicit_formulas} \linebreak \noindent\hyperlink{gauged_paths_of_connections}{Gauged paths of connections}\dotfill \pageref*{gauged_paths_of_connections} \linebreak \noindent\hyperlink{InInfCSTheory}{In $\infty$-Chern-Weil theory}\dotfill \pageref*{InInfCSTheory} \linebreak \noindent\hyperlink{PrerequisitesHigher}{Prerequisites}\dotfill \pageref*{PrerequisitesHigher} \linebreak \noindent\hyperlink{HigherOrderChernSimonsForms}{Higher order Chern-Simons form}\dotfill \pageref*{HigherOrderChernSimonsForms} \linebreak \noindent\hyperlink{OrdinaryCSRevisited}{Ordinary Chern-Simons forms revisited}\dotfill \pageref*{OrdinaryCSRevisited} \linebreak \noindent\hyperlink{as_secondary_characteristic_forms}{As secondary characteristic forms}\dotfill \pageref*{as_secondary_characteristic_forms} \linebreak \noindent\hyperlink{chernsimons_theory}{Chern-Simons theory}\dotfill \pageref*{chernsimons_theory} \linebreak \noindent\hyperlink{in_terms_of_lie_algebroids}{In terms of $\infty$-Lie algebroids}\dotfill \pageref*{in_terms_of_lie_algebroids} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{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]]: \begin{displaymath} d_{dR} CS(A) = \langle F_A \wedge \cdots \wedge F_A \rangle \,, \end{displaymath} 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 \emph{difference} between the two curvature characteristic forms \begin{displaymath} d_{dR}CS(A,A') = \langle (F_A)^k \rangle - \langle (F_{A'})^k \rangle \,. \end{displaymath} Chern-Simons forms are of interest notably when the differential forms $A,A'$ are (local representatives of) [[connection on a bundle|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 \emph{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 \begin{displaymath} \langle A \wedge d_{dR} A\rangle + c \langle A \wedge [A \wedge A] \rangle \,. \end{displaymath} 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 [[trace]]s: $\langle \cdots \rangle = tr(\cdots )$. \hypertarget{Details}{}\subsection*{{Details}}\label{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. \hypertarget{PathsOfConnections}{}\subsubsection*{{Paths of connections}}\label{PathsOfConnections} Let $U$ be a [[smooth manifold]]. \begin{prop} \label{}\hypertarget{}{} A \textbf{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 \textbf{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 \begin{displaymath} \psi_t : U \simeq U \times * \stackrel{Id \times t}{\to} U \times [0,1] \end{displaymath} picks the copy of $U$ at parameter $t$. \end{prop} So a smooth path is a smooth 1-form on the cylinder $U \times [0,1]$ and it is \emph{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. \begin{defn} \label{}\hypertarget{}{} Let $P$ be an [[invariant polynomial]] on $\mathfrak{g}$ of arity $n$. Consider the [[fiber integration]] \begin{displaymath} CS_P(A_0,A_1) := \int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \,. \end{displaymath} 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 form]]s of $A_0$ and $A_1$: \begin{displaymath} 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}) \end{displaymath} \end{defn} \begin{proof} Write the fiber integration more explicitly as an [[integral]] \begin{displaymath} 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 \,. \end{displaymath} 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{displaymath} \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} \,. \end{displaymath} \end{proof} \hypertarget{explicit_formulas}{}\subsubsection*{{Explicit formulas}}\label{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 \begin{displaymath} CS_P(A_0, A_1) = \int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \,. \end{displaymath} 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 \begin{displaymath} F_{\hat A}(t) = F_{A_t} + (\partial_t A_t) \wedge d t \,. \end{displaymath} Inserting this into the above expression yields \begin{displaymath} CS_P(A_0,A_1) = \int_0^1 P(\partial_t A \wedge F_{A_t} \wedge \cdots \wedge F_{A_t}) \,. \end{displaymath} Notably if $A_0 = 0$ and $\hat A$ is the \emph{constant} path $\hat A : t \mapsto t A$ to $A_1 := A$ such that \begin{displaymath} \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} \end{displaymath} this yields \begin{displaymath} 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]))) \,. \end{displaymath} 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 \begin{displaymath} 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 \,. \end{displaymath} Particularly for $n = 2$ and using the definition of the [[curvature]] 2-form $F_A = d_{dR} A + [A \wedge A]$ we get \begin{displaymath} CS(A) = \langle A \wedge d A\rangle + c \langle A \wedge [A \wedge A]\rangle \,. \end{displaymath} \hypertarget{gauged_paths_of_connections}{}\subsubsection*{{Gauged paths of connections}}\label{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 \emph{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 \begin{displaymath} \hat A : t \mapsto A_t + \lambda d t \,, \end{displaymath} 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 \begin{displaymath} F_{\hat A} : t \mapsto F_{A_t} + ((\partial_t A)_t + d_U \lambda(t) + [\lambda,A_t]) \wedge d t \end{displaymath} \begin{prop} \label{}\hypertarget{}{} 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. \end{prop} \hypertarget{InInfCSTheory}{}\subsection*{{In $\infty$-Chern-Weil theory}}\label{InInfCSTheory} We discuss now a more encompassing perspective on Chern-Simons forms the way it occurs in [[∞-Chern-Weil theory]]. \hypertarget{PrerequisitesHigher}{}\subsubsection*{{Prerequisites}}\label{PrerequisitesHigher} 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-algebra]]s naturally associated with it: \begin{itemize}% \item the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$; \item the [[Weil algebra]] $W(\mathfrak{g})$; \item the algebra of [[invariant polynomial]]s $inv(\mathfrak{g})$. \end{itemize} 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]] \begin{displaymath} \exp(\mathfrak{g}) : U,[n] \mapsto dgAlg( CE(\mathfrak{g}), C^\infty(U)\otimes \Omega^\bullet(\Delta^n) ) \,, \end{displaymath} where here and in the following $\Omega^\bullet(\Delta^n)$ denotes the [[de Rham complex]] dg-algebra of those smooth [[differential form]]s $\omega$ on the standard smooth $n$-simplex that have \emph{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 \begin{displaymath} \mathbf{B}G = \mathbf{cosk}_{n+1} \exp(\mathfrak{g}) \,, \end{displaymath} 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 form]]s in that it encodes [[pseudo-connection]]s on $G$-[[principal ∞-bundle]]s is given by the simplicial presheaf \begin{displaymath} \mathbf{B}G_{diff} := \mathbf{cosk}_{n+1} \left( U,[n] \mapsto \left\{ \itexarray{ 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) \,, \end{displaymath} 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 \begin{displaymath} CE(\mathfrak{g}) \leftarrow CE(b^{k-1} \mathbb{R}) : \mu \,. \end{displaymath} Simply by composition (since we ignore the truncation for the moment), this integrates to a cocycle of the corresponding $\infty$-Lie groupoids \begin{displaymath} \exp(\mu) : \exp(\mathfrak{g}) \to \exp(b^{k-1}\mathbb{R}) \,, \end{displaymath} At [[∞-Chern-Weil theory]] it is discussed how the proper lift of this through the extension $\mathbf{B}G_{diff}$ that computes the [[schreiber:differential cohomology in an (∞,1)-topos|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 \begin{displaymath} \itexarray{ 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}) } \end{displaymath} with a choice of interpolating Chern-Simons element $cs \in W(\mathfrak{g})$, which induces by precomposition with its upper part the morphism \begin{displaymath} \exp((cs,\langle-,-\rangle)) : \mathbf{B}G_{diff} \to \exp(b^{k-1}\mathbb{R})_{diff} \,. \end{displaymath} By further projection to its lower part we get furthermore a morphism \begin{displaymath} \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}) \}) \,. \end{displaymath} 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 (see also [[circle n-bundles with connection]] the section ) 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 \begin{displaymath} \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}(-)) \end{displaymath} to the image of the $\mathbb{R}$-[[Deligne complex]] of sheaves under the [[Dold-Kan correspondence]]. \hypertarget{HigherOrderChernSimonsForms}{}\subsubsection*{{Higher order Chern-Simons form}}\label{HigherOrderChernSimonsForms} 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 \begin{displaymath} A : X \to \mathbf{B}G_{diff} \end{displaymath} 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 \begin{displaymath} X \to \mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn} \end{displaymath} discussed above produces the corresponding [[curvature characteristic form]]. A [[homotopy]] \begin{displaymath} (\nabla \to \nabla') : X \cdot \Delta[1] \to \mathbf{B}G_{diff} \end{displaymath} is a smooth path in the space of $\mathfrak{g}$-valued forms on $X$. Under the [[adjunction]] \begin{displaymath} [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]}] \end{displaymath} this corresponds to a $(k-1)$-form on $X$ this is the Chern-Simons form \begin{displaymath} CS(\nabla \to \nabla') : X \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}^{\Delta[1]} \,. \end{displaymath} The higher homotopies are higher order Chern-Simons forms. The following proposition says this in a more precise way for ordinary Chern-Simons forms. \hypertarget{OrdinaryCSRevisited}{}\paragraph*{{Ordinary Chern-Simons forms revisited}}\label{OrdinaryCSRevisited} We now show how the \hyperlink{PathsOfConnections}{traditional definition} of Chern-Simons forms is reproduced by the \hyperlink{HigherOrderChernSimonsForms}{general abstract} mechanism. \begin{prop} \label{}\hypertarget{}{} \textbf{(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) \begin{displaymath} A : X \to \mathbf{B}G_{diff} \end{displaymath} are in canonical bijection with [[Lie-algebra valued 1-forms]] $A \in \Omega^1(X,\mathfrak{g})$. Morphisms \begin{displaymath} X \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn} \end{displaymath} are in canonical bijection with closed $k$-forms on $X$ and composition with the morphism \begin{displaymath} \mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn} \end{displaymath} discussed \hyperlink{PrerequisitesHigher}{above} and under this canonical identification the composite \begin{displaymath} \langle F_A \rangle : X \stackrel{A}{\to} \mathbf{B}G_{diff} \stackrel{}{\to} \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn} \end{displaymath} is the corresponding [[curvature characteristic form]]. Homotopies \begin{displaymath} (A_1 \stackrel{\gamma}{\to} A_2) : X\times \Delta[1] \to \mathbf{B}G_{diff} \end{displaymath} 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 \begin{displaymath} \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]} \,. \end{displaymath} \end{prop} \begin{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 \begin{displaymath} \itexarray{ 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}) } \,. \end{displaymath} 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 \begin{displaymath} \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 \end{displaymath} 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 \begin{displaymath} \itexarray{ 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}) } \,. \end{displaymath} 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]] [[Lie integration|integrating]]] $\mathfrak{g}$ based at the identity, $g(x,0) = e$, by the formula \begin{displaymath} \lambda = g^* \theta \end{displaymath} where $\theta \in \Omega^1(G, \mathfrak{g})$ is the [[Maurer-Cartan form]] on $G$, or conversely by [[parallel transport]] \begin{displaymath} f(x,s) = P \exp(\int_{[0,s]} \lambda(x,s) d s) \end{displaymath} We may think of this as a smooth \emph{path of gauge transformations} . The bottom morphism encodes a $\mathfrak{g}$-valued form \begin{displaymath} \hat A + \lambda \in \Omega^1(X \times [0,1] , \mathfrak{g}) \end{displaymath} 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 \emph{smooth path in the space of 1-forms} . In the case that $\lambda = 0$ this is a \emph{pure shift path} in the \hyperlink{PathsOfConnections}{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 \begin{displaymath} \langle F_{\hat A } \rangle \in \Omega^{k}(X)\otimes C^\infty(\Delta^1)\oplus \Omega^{k-1}(X)\otimes \Omega^1(\Delta^1) \,. \end{displaymath} The [[fiber integration]] of this over $\Delta^1$ is manifestly the same operation as that in the definition of the Chern-Simons form \hyperlink{PathsOfConnections}{above}. \end{proof} \hypertarget{as_secondary_characteristic_forms}{}\subsection*{{As secondary characteristic forms}}\label{as_secondary_characteristic_forms} If a [[curvature characteristic form]] \emph{vanishes} (for instance if the [[connection on a bundle|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 \textbf{[[secondary characteristic class]]}. \hypertarget{chernsimons_theory}{}\subsubsection*{{Chern-Simons theory}}\label{chernsimons_theory} In particular on a 3-dimensional [[smooth manifold]] $X$ necessarily the Chern-Simons 3-form is closed. The [[functional]] \begin{displaymath} (A \in \Omega^1(X,\mathfrak{g})) \mapsto \int_X CS(A) \end{displaymath} 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. \hypertarget{in_terms_of_lie_algebroids}{}\subsection*{{In terms of $\infty$-Lie algebroids}}\label{in_terms_of_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 \begin{displaymath} \itexarray{ 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) } \end{displaymath} the data of an [[Ehresmann connection]] on a $G$-[[principal bundle]] expressed as a diagram of [[∞-Lie algebroid]]s with the [[curvature characteristic form]]s 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 \begin{displaymath} \itexarray{ \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) } \,. \end{displaymath} 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]] $P(F_A)$ \begin{displaymath} \itexarray{ 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) } \,. \end{displaymath} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[∞-Lie algebroid valued differential forms]] \item [[connection on an ∞-bundle]] \item [[curvature]] \item [[Bianchi identity]] \item [[curvature characteristic form]] \item \textbf{Chern-Simons form} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The article introducing the concept is \begin{itemize}% \item [[Shiing-Shen Chern]], [[James Simons]], \emph{[[Characteristic forms and geometric invariants]]} The Annals of Mathematics, Second Series, Vol. 99, No. 1 (Jan., 1974), pp. 48-69 (\href{http://www.jstor.org/pss/1971013}{jstor}) (\href{http://www.dpmms.cam.ac.uk/~hk244/chern-simons.cp.pdf}{pdf}) \end{itemize} 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 [[orthogonal group|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 \begin{itemize}% \item [[Jean-Luc Brylinski]], [[Dennis McLaughlin]] \emph{ech cocycles for characteristic classes} , Comm. Math. Phys. 178 (1996) () \end{itemize} The statements about ``pure shift'' paths are reviewed on the first few pages of \begin{itemize}% \item [[James Simons]], [[Dennis Sullivan]], \emph{Structured vector bundles define differential K-theory} (\href{http://arxiv.org/abs/0810.4935?context=math.DG}{arXiv}) \end{itemize} which discusses the relevance of Chern-Simons forms in [[differential K-theory]]. The [[L-∞-algebra]]-formulation is discussed in \href{http://arxiv.org/abs/0801.3480}{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 \begin{itemize}% \item Israel M. Gelfand, Mikhail M. Smirnov, \emph{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. \textbf{123}, Birkh\"a{}user 1994; \emph{Chern-Simons classes and cocycles on the Lie algebra of the gauge group}, The Gelfand Mathematical Seminars, 1993--1995, 101--122, Birkh\"a{}user 1996. \end{itemize} [[!redirects Chern-Simons forms]] [[!redirects Chern-Simons 3-form]] [[!redirects Chern-Simons 3-forms]] \end{document}