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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*{universal Chern-Simons line 3-bundle} [[!redirects universal Chern-Simons circle 3-bundle]] [[!redirects Chern-Simons circle 3-bundle]] \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{constructions}{Constructions}\dotfill \pageref*{constructions} \linebreak \noindent\hyperlink{in_differential_geometry__for_lie_groups}{In differential geometry -- For Lie groups}\dotfill \pageref*{in_differential_geometry__for_lie_groups} \linebreak \noindent\hyperlink{InCechDelineCohomology}{In Cech-Deligne cohomology}\dotfill \pageref*{InCechDelineCohomology} \linebreak \noindent\hyperlink{InInfChernWeil}{In $\infty$-Chern-Weil theory}\dotfill \pageref*{InInfChernWeil} \linebreak \noindent\hyperlink{as_a_bundle_2gerbe}{As a bundle 2-gerbe}\dotfill \pageref*{as_a_bundle_2gerbe} \linebreak \noindent\hyperlink{in_complex_analytic_geometry__for_complex_lie_groups}{In complex analytic geometry -- For complex Lie groups}\dotfill \pageref*{in_complex_analytic_geometry__for_complex_lie_groups} \linebreak \noindent\hyperlink{ForReductiveAlgebraicGroups}{In arithmetic geometry -- For reductive groups}\dotfill \pageref*{ForReductiveAlgebraicGroups} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{for_lie_groups}{For Lie groups}\dotfill \pageref*{for_lie_groups} \linebreak \noindent\hyperlink{the_cechdeligne_cohomology_realization}{The Cech-Deligne cohomology realization}\dotfill \pageref*{the_cechdeligne_cohomology_realization} \linebreak \noindent\hyperlink{2GerbeReferences}{The 2-gerbe realization}\dotfill \pageref*{2GerbeReferences} \linebreak \noindent\hyperlink{ReferencesForComplexReductiveGroups}{For complex reductive groups}\dotfill \pageref*{ReferencesForComplexReductiveGroups} \linebreak \noindent\hyperlink{for_reductive_algebraic_groups}{For reductive algebraic groups}\dotfill \pageref*{for_reductive_algebraic_groups} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{Chern-Simons circle 3-bundle} is the [[circle n-bundle with connection]] classified by the cocycle in degree 4 [[ordinary differential cohomology]] that is canonically associated to a $G$-[[principal bundle]] with [[connection on a bundle|connection]]. For $G$ a [[Lie group]] and $c : \mathcal{B}G \to K(\mathbb{Z},4)$ a cocycle for a degree 4 [[characteristic class]] in [[integral cohomology]] and $X$ a [[smooth manifold]], [[Chern-Weil theory]] provides a morphism (the refined [[Chern-Weil homomorphism]]) \begin{displaymath} \hat c : G Bund_\nabla(X) \to H_{diff}^4(X) \end{displaymath} from $G$-[[principal bundle]]s with [[connection on a bundle|connection]] $\nabla$ to degree 4 [[ordinary differential cohomology]]. The cocycles on the right may be thought of as \begin{itemize}% \item [[circle n-bundle with connection|circle 3-bundles with connection]] $\hat c(\nabla)$; \item [[bundle gerbe|bundle 2-gerbe]]s with connection. \end{itemize} By construction, the [[curvature]] 4-form of $\hat c(\nabla)$ is the [[curvature characteristic form]] $\langle F_\nabla \wedge F_\nabla\rangle$ of $\nabla$ and accordingly the 3-form connection on $\hat c(\nabla)$ is locally a [[Chern-Simons form]] $CS(\nabla)$ of $\nabla$. Accordingly, the [[higher parallel transport]] induced by $\hat c(\nabla)$ over 3-dimensional manifolds $\phi : \Sigma \to X$ is the [[action functional]] of the [[quantum field theory]] called [[Chern-Simons theory]]. In this form it appears for instance as the [[gauge field]] called the [[supergravity C-field]] in certain [[supergravity]] theories. In particular, if (with due care) one takes $\nabla$ to be the \emph{universal connection on the $G$-[[universal principal bundle]]} over a smooth version of $B G$, then $\hat c(\nabla)$ is the background [[gauge field]] for bare [[Chern-Simons theory]]. Therefore this structure $\hat c(\nabla)$ has become known as the \textbf{Chern-Simons 2-gerbe} of $\nabla$. We may also think of it as the \emph{Chern-Simons [[circle n-bundle with connection|circle 3-bundle]]} . At least for simply connected $G$ one may enhance the assignment $\nabla \mapsto \hat c(\nabla)$ to a morphism of [[∞-groupoid]]s \begin{displaymath} \hat \mathbf{c} : \mathbf{H}_conn(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1)) \,, \end{displaymath} where on the left we have the [[groupoid]] of smooth $G$-principal bundles with connection on $X$, and on the right the 3-groupoid of circle 3-bundles with connection. The [[homotopy fiber]]s of this morphism over a trivial circle 3-bundle with connection are 3-groupoids whose objects are naturally identified with pairs consisting of a connection $\nabla$ on a $G$-bundle and a \emph{trivialization} of its corresponding Chern-Simons 3-bundle. This in particular implies a trivialization of the underlying cocycle in degree 4 [[integral cohomology]] and therefore defines a [[string structure]]. One calls these homotopy fibers therefore [[differential string structure]]s. \hypertarget{constructions}{}\subsection*{{Constructions}}\label{constructions} \hypertarget{in_differential_geometry__for_lie_groups}{}\subsubsection*{{In differential geometry -- For Lie groups}}\label{in_differential_geometry__for_lie_groups} \hypertarget{InCechDelineCohomology}{}\paragraph*{{In Cech-Deligne cohomology}}\label{InCechDelineCohomology} In (\hyperlink{GeomConstructionFirst}{Brylinski-McLaughlin I}) there is spelled out an explicit construction of $\hat c(\nabla)$ for given $\nabla$ in [[Cech cohomology|Cech]]-[[Deligne cohomology]]. This is a special case of the general construction presented in (\hyperlink{CechCocyclesForCharClasses}{Brylinski-McLaughlin II}). In this section here we review this explicit cocycle construction. In the \hyperlink{InInfChernWeil}{next section} we discuss a systematic way to derive this construction. Assume that $G$ is a [[simply connected]] compact simple [[Lie group]], such as the [[spin group]], and take the characteristic class $c$ to be that whose [[transgression]] to $G$ has as image in [[de Rham cohomology]] the de Rham class of the normalized canonical [[Lie algebra cohomology|Lie algebra cocycle]] $\mu \in CE(\mathfrak{g})$. For $P \to X$ a $G$-bundle with connection $\nabla$, there exists an [[open cover]] $\{U_i \to X\}$ such that we have a [[Cech cohomology]] cocycle for $P$ given by a smooth transition function \begin{displaymath} g_{i,j} : U_i \cap U_j \to G \end{displaymath} satisfying on $U_i \cap U_j \cap U_k$ the cocycle condition $g_{i j} \cdot g_{j k} = g_{i k}$. Since $G$ is assumed connected and [[simply connected]] and since for every Lie group the second [[homotopy group]] is trivial we have that the first nonvanishing homotopy group of $G$ is the third one. Therefore we can always find (possibly after refining the cover) a lift of this cocycle as follows: \begin{itemize}% \item on double intersections, choose smooth functions \begin{displaymath} \hat g_{i j} : (U_i \cap U_j) \times \Delta^1 \to G \end{displaymath} such that $\hat g_{i j}(x,0 ) = e$ is the identity in $g$, and such that such that $\hat g_{i j}(x,1) = g_{i j}(x)$; \item on triple intersections, choose smooth functions \begin{displaymath} \hat g_{i j k} : (U_i \cap U_j \cap U_k) \times \Delta^2 \to G \end{displaymath} that cobound these paths in the evident way \begin{displaymath} \itexarray{ && g_{i j} \\ & {}^{\mathllap{\hat g_{i j}}}\nearrow &\Downarrow^{\mathrlap{\hat g_{i j k}}}& \searrow^{g_{i j} \cdot \mathrlap{\hat g_{j k}}} \\ e &&\underset{\hat g_{i k}}{\to}&& g_{i k} }. \end{displaymath} (This can be done because $\pi_1(G) = *$.) \item on quadruple intersection choose smooth functions \begin{displaymath} \hat g_{i j k l} : (U_i \cap U_j \cap U_k \cap U_l) \times \Delta^3 \to G \end{displaymath} such that these 3-balls fill the evident tetrahedra. (This can be done because $\pi_2(G) = 0$.) \end{itemize} \begin{prop} \label{}\hypertarget{}{} The [[Cech cohomology]] cocycle with coefficients in $\mathbf{B}^3 \mathbb{R}/\mathbb{Z}$ which is given by \begin{displaymath} c(g)_{i j k l} := \int_{\Delta^3} \hat g_{i j k l}^* \mu \;\;\;\;\; mod \mathbb{Z} \end{displaymath} is well defined and represents the class $c(P) \in H^4(X)$ in [[integral cohomology]]. Moreover, this refines to the cocycle in Cech-[[Deligne cohomology]] that is given by \begin{displaymath} \left( CS(\sigma_i^* A) \,,\;\; \int_{\Delta^1} CS((P\cdot \hat g_{i j})^* A) \,,\;\; \int_{\Delta^2} CS((P\cdot \hat g_{i j k})^* A) \,,\;\; \int_{\Delta^3} \hat g_{i j k l}^* \mu \;\;\; mod \mathbb{Z} \right) \,, \end{displaymath} where \begin{itemize}% \item $A \in \Omega^1(P,\mathfrak{g})$ is the incarnation of the connection $\nabla$ as an [[Ehresmann connection]] given by a glbally defined 1-form on the total space $P \to X$ of the bundle; \item $\sigma_i : U_i \to P$ are the local [[section]]s of $P \to X$ that induce the original Cech cocycle $(g_{i j} := \sigma_i \cdot \sigma_j^{-1})$; \item $P \cdot \hat g_{i j} : \Delta^1 \to P$ is given by the right [[action]] of $G$ on $P$ and analogously for the other terms; \item $CS(...)$ denotes the [[Chern-Simons form]] of the given $\mathfrak{g}$-valued 1-form. \end{itemize} \end{prop} \begin{proof} First notice that this is indeed well-defined: by compactness and simplicty of $G$ we have $\pi_3(G) = \mathbb{Z}$. By assumption on $\mu \in \Omega^3(G)$, for any map $f : S^3 \to G$, we have $\int_{S^3} f^*\mu \in \mathbb{Z} \subset \mathbb{R}$. This implies that $c(g)$ is indeed a Cech cocycle. Then the proof is effectively just the observation that the given collection of differential forms indeed does refine this to a Cech-cocycle with coefficients in the Deligne complex, and that therefore we can read off the image of the integral cohomology class $[c(g)]$ in de Rham cohomology from the curvature 4-form of this Deligne cocycle. That is by construction $\langle F_\nabla \wedge F_\nabla \rangle \in \Omega^4_{cl}(X)$, which by [[Chern-Weil theory]] is indeed the image of the claimed integral class. \end{proof} So the only mystery about this construction is really: where does it come from? Apart from making this clever Ansatz and checking that it works, can one somehow systematically derive this construction? This we shall try to answer the section \hyperlink{InInfChernWeil}{below}. \hypertarget{InInfChernWeil}{}\paragraph*{{In $\infty$-Chern-Weil theory}}\label{InInfChernWeil} The above Cech-Deligne cocycle construction of $\hat c(\nabla)$ may be understood as a special case of the general construction of Chern-Weil homomorphisms by the methods discussed at [[∞-Chern-Weil theory]]. We briefly recall the general approach and then spell out the details. The basic ingredients in [[∞-Chern-Weil theory]] that give the refinement of a [[characteristic class]] to a morphism \begin{displaymath} \mathbf{H}_{conn}(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X, \mathbf{B}^k U(1)) \end{displaymath} from the $G$-principal bundles with connection to [[ordinary differential cohomology]] are these: \begin{enumerate}% \item for a given [[Lie algebra]] $\mathfrak{g}$ the realization of the corresponding [[Lie group]] as a truncation of the simplicial presheaf \begin{displaymath} (U,[n]) \mapsto \{\Omega^\bullet(U\times \Delta^n) \leftarrow CE(\mathfrak{g})\} \end{displaymath} (see [[Lie integration]]); \item the observation that, up to subtleties with the truncation, a [[Lie algebra cohomology|Lie algebra cocycle]] \begin{displaymath} CE(\mathfrak{g}) \leftarrow CE(b^{k-1}\mathbb{R}) : \mu \end{displaymath} induces therefore an integrated cocycle $\mathbf{B}G \to \mathbf{B}^k U(1)$; \item the observation that this is lifted to connections and differential refinement by \begin{enumerate}% \item thickening the simplicial presheaf for $\mathbf{B}G$ to \begin{displaymath} (U,[n]) \mapsto \left\{ \itexarray{ C^\infty(U) \otimes \Omega^\bullet(\Delta^n) &\stackrel{}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right\} \end{displaymath} \item thickening the Lie algebra cocycle by its Chern-Simons element \end{enumerate} \end{enumerate} \begin{displaymath} \itexarray{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) } \end{displaymath} and then postcomposing with that. Note that the above diagram is part of a larger diagram involving the invariant polynomial $\langle-\rangle$ for $\mu$ and exhibiting the Chern-Simons element as a transgression element between these two: \begin{displaymath} \itexarray{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) } \end{displaymath} Also note that $inv(b^{k-1} \mathbb{R})\cong CE(b^k \mathbb{R})$. For the case at hand, let $\mathfrak{g}$ be a [[semisimple Lie algebra]], $\langle -\rangle : CE(b^3\mathbb{R})\to W(\mathfrak{g})$ its canonical [[Killing form]] [[invariant polynomial]], $\mu = \langle -,[-,-]\rangle: CE(b^2\mathbb{R})\to CE(\mathfrak{g})$ the corresponding [[Lie algebra cohomology|Lie algebra cocycle]], $cs: W(b^2\mathbb{R})\to W(\mathfrak{g})$ the Chern-Simons elements exhibiting the transgression between the two, $G$ the [[simply connected]] [[Lie group]] [[Lie integration|integrating]] it. First consider the bare cocycle for the Chern-Simons circle 3-bundle as the $\mu$. \begin{def} \label{}\hypertarget{}{} Consider the [[simplicial presheaf]] \begin{displaymath} \exp(\mathfrak{g}) : (U,[n]) \mapsto \{C^\infty(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow CE(\mathfrak{g})\} \,, \end{displaymath} where here and in what follows differential forms $\omega$ on simplices are taken to have \emph{sitting instants} in that for all $k \in \mathbb{N}$ there exists for every $k$-face of $\Delta^n$ an open neighbourhood such that $\omega$ restricted to that open neighbourhood is constant in the direction perpendicular to the boundary. \end{def} \begin{lemma} \label{}\hypertarget{}{} The canonical map \begin{displaymath} \mathbf{cosk}_3 \exp(\mathfrak{g}) \to \mathbf{B}G \end{displaymath} from the 3-[[coskeleton]] of $\exp(\mathfrak{g})$ to the [[delooping]] of the simply connected Lie group $G$ which is given on 1-morphisms by [[higher parallel transport]] is an equivalence ( in the [[model structure on simplicial presheaves]] $[CartSp^{op}, sSet]_{proj}$). \end{lemma} \begin{proof} Use that a $\mathfrak{g}$-valued 1-form on the interval is canonically identified with a based path in $G$. Then use that for $k \leq 2$ we have $\pi_k(G) = 0$. See [[Lie integration]] for more. \end{proof} \begin{prop} \label{}\hypertarget{}{} There is a commuting diagram \begin{displaymath} \itexarray{ \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^2 \mathbb{R}) \\ \downarrow && \downarrow^{\mathrlap{\int_{\Delta^\bullet}}} \\ \mathbf{cosk}_3 \exp(\mathfrak{g}) &\stackrel{}{\to}& \mathbf{B}^3 \mathbb{R}/\mathbb{Z} } \,, \end{displaymath} where the right vertical morphism is the composite of the equivalence \begin{displaymath} \int_{\Delta^\bullet} : \exp(b^2 \mathbb{R}) \stackrel{\simeq}{\to} \mathbf{B}^3 \mathbb{R} \end{displaymath} discussed at with the evident quotient $\mathbf{B}^3 \mathbb{R} \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}$, where the copy of $\mathbb{Z}$ in $\mathbb{R}$ is the lattice of periods of $\mu$ over 3-spheres in $G$. \end{prop} The top morphism sends a $U$-family of 3-morphisms $\Omega^\bullet(U \times \Delta^3) \stackrel{A}{\leftarrow} CE(\mathfrak{g})$ -- which we may think of as a $U$-family of based 3-balls $\Sigma : U \times \Delta^3 \to G$ -- to the family of 3-forms \begin{displaymath} \Omega^\bullet(U \times \Delta^k)_{vert} \stackrel{A}{\leftarrow} CE(\mathfrak{g}) \stackrel{\mu}{\leftarrow} CE(b^2 \mathbb{R}) : \mu^*(A) \,. \end{displaymath} which we may think as a family of closed 3-forms \begin{displaymath} \mu^*(A)\in \Omega^\bullet(U \times \Delta^k)_{vert} \end{displaymath} The right vertical morphism sends this to the [[fiber integration]] \begin{displaymath} U \mapsto \int_{\Delta^3} \mu^*(A) \;\;\; \in C^\infty(U;\mathbb{R}) \end{displaymath} and regards the result then modulo $\mathbb{Z}$. That this indeed gives a morphism down at the bottom is the statement that for a 4-morphism in $\mathbf{cosk}_3 \exp(\mathfrak{g})$ -- which is a 3-sphere $V : S^3 \to G$ -- we have that $\int_{S^3} V^* \mu^*(A) = 0 \;mod\; \mathbb{Z}$, which is true by the fact that we take $\mathbb{Z}$ to be precisely generated by these periods. (Alternatively we can assume $\mu$ to be normalized such that it generates the image in deRham cohomology of $H^3(G,\mathbb{Z}) \simeq \mathbb{Z}$.) We shall by slight abuse of notation write $\exp(\mu)$ also for the morphism $\mathbf{cosk}_3 \exp(\mathfrak{g}) \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}$. \begin{lemma} \label{}\hypertarget{}{} For $\{U_i \to X\}$ a [[cover]] and $C(U) \in [CartSp^{op}, sSet]_{proj}$ the corresponding [[Cech nerve]] we have that \begin{itemize}% \item a morphism $g : C(U) \to \mathbf{B}G$ is precisely a Cech 1-cocycle with values in $G$; \item a lift \begin{displaymath} \itexarray{ && \mathbf{cosk}_3 \exp(\mathfrak{g}) \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}} \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G } \end{displaymath} is precisely a lift of this cocycle to a system of paths, triangles and tetrahedra in $G$, as \hyperlink{InCechDelineCohomology}{above}. \end{itemize} \end{lemma} \begin{def} \label{}\hypertarget{}{} Write $\exp(\mathfrak{g})_{diff}$ for the [[simplicial presheaf]] \begin{displaymath} (U,[n]) \mapsto \left\{ \itexarray{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\leftarrow& W(\mathfrak{g}) } \right\} \,. \end{displaymath} \end{def} Its 3-[[coskeleton]] $\mathbf{cosk}_3 \exp(\mathfrak{g} \to inn(\mathfrak{g}))$ is the coefficient for $G$-principal bundles with [[pseudo-connection]] adapted to the model $\mathbf{cosk}_3 \exp(\mathfrak{g})$ for $\mathbf{B}G$. \begin{lemma} \label{}\hypertarget{}{} Pseudo-connections $\hat \nabla$ \begin{displaymath} \itexarray{ && \mathbf{cosk}_3 \exp(\mathfrak{g})_{diff} \\ &{}^{\mathllap{\hat \nabla}}\nearrow& \downarrow^{\mathrlap{\simeq}} \\ && \mathbf{cosk}_3 \exp(\mathfrak{g}) \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}} \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G } \end{displaymath} which are \emph{genuine} $\infty$-connections in that their curvature components have no leg along the simplicial directions are in bijection with ordinary connections on the $G$-bundle given by $C(U) \to \mathbf{B}G$. \end{lemma} \begin{proof} On single patches $\hat \nabla$ is a collection of $\mathfrak{g}$-valued 1-forms $A_i \in \Omega^1(U_i, \mathfrak{g})$. On double intersection it is a collection \begin{displaymath} \hat A_{i j} = A_{i j} + \lambda_{i j} \in \Omega^1(U_i \cap U_j, \mathfrak{g})\otimes C^\infty(\Delta^1) \oplus C^\infty(U)\otimes \Omega^1(\Delta^1, \mathfrak{g}) \subset \Omega^1(U_i \cap U_j \times \Delta^1, \mathfrak{g}) \end{displaymath} whose restriction $\lambda_{i j}$ to $\Delta^1$ is the given path $\hat g_{i j}$ that is being covered. The condition that the curvature of $\hat A_{i j}$ has no component in the simplicial direction is the [[differential equation]] \begin{displaymath} \frac{\partial}{\partial t} A_{i j} = d_U (\lambda_{i j})_t + [(\lambda_{i j})_t, A_{i j}] \,. \end{displaymath} This differential equation has a unique solution for the boundary condition $A_{i j}(0) = A_i$ given by \begin{displaymath} A_{i j}(t) = \hat g_{i j}(t)^{-1}(A_i + d)\hat g_{i j}(t) \,. \end{displaymath} (To see this, use the formulas from [[parallel transport]]. If we assume just for notational simplicity that we are dealing with a [[matrix Lie algebra]] then we have $\frac{\partial}{\partial t} \hat g_{i j} = \hat g_{i j} \cdot \lambda$ (by definition) and using that the claim follows.) In particular this implies the forms on single patches satisfy the ordinary cocycle relation \begin{displaymath} A_{i j}(1) = A_j = g_{i j}^{-1}(A_i + d)g_{i j} \end{displaymath} for connections. Similarly there are differential equations on 2-simplices and 3-simplices with unique solutions. \end{proof} \begin{lemma} \label{}\hypertarget{}{} Pasting postcomposition with the diagram \begin{displaymath} \itexarray{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^2 \mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs,\langle \rangle)}{\leftarrow}& W(b^2 \mathbb{R}) } \end{displaymath} induces a morphism $\exp(\mathfrak{g})_{diff} \to \exp(b^2\mathbb{R})_{diff}$ and we obtain a commuting diagram \begin{displaymath} \itexarray{ \exp(\mathfrak{g})_{diff} &\to& \exp(b^2\mathbb{R})_{diff} \\ \downarrow && \downarrow \\ \mathbf{cosk}_3\exp(\mathfrak{g})_{diff} &\to& \mathbf{B}^3 U(1)_{chn,diff} } \end{displaymath} that covers the corresponding diagram we had before. \end{lemma} Here we are using the object $\mathbf{B}^3 U(1)_{ch,diff}$ described in detail at [[circle n-bundle with connection]]. \begin{remark} \label{}\hypertarget{}{} The deeper reason for this construction is that the zig-zag composite \begin{displaymath} \mathbf{B}G \stackrel{\simeq}{\leftarrow} \mathbf{cosk}_3\exp(\mathfrak{g})_{diff} \to \mathbf{B}^3 U(1)_{diff,chn} \to \mathbf{\flat}_{dR}\mathbf{B}^4 U(1)_{chn} \end{displaymath} of morphisms of simplicial presheaves models the intrinsically defined morphism \begin{displaymath} \mathbf{B}G \to \mathbf{\flat}_{dR}\mathbf{B}^4 U(1) \end{displaymath} in the [[(∞,1)-topos]] [[?LieGrpd]]. \end{remark} \begin{prop} \label{}\hypertarget{}{} The outer composite morphism \begin{displaymath} \itexarray{ && \mathbf{cosk}_3 \exp(\mathfrak{g})_{diff} &\stackrel{\exp((cs,\langle -\rangle))}{\to}& \mathbf{B}^3 U(1)_{diff} \\ &{}^{\mathllap{\hat \nabla}}\nearrow& \downarrow^{\mathrlap{\simeq}} \\ && \mathbf{cosk}_3 \exp(\mathfrak{g}) \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}} \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G } \end{displaymath} is precisely the Cech-Deligne cocycle \begin{displaymath} (CS(A_i), \int_{\Delta^1} CS(\hat A_{i j} ), \int_{\Delta^2} CS(\hat A_{i j k}), \int_{\Delta^3} \mu(A_{i j k l})) \,. \end{displaymath} This is is exactly equal to the cocycle discussed \hyperlink{InCechDelineCohomology}{above}. \end{prop} Notice by the way that this construction also serves as a manifest proof that this collection of data indeed does constitute a Deligne cocycle. \begin{proof} This is a matter of plugging the above pieces into each other. For instance, on double intersections we have that the 3-form $CS(\hat A_{i j})$ is the image of the degree 3-generator on $W(b^2 \mathbb{R})$ under the composite \begin{displaymath} \Omega^\bullet(U \times \Delta^k)_{vert} \stackrel{\hat A_{i j}}{\leftarrow} W(\mathfrak{g}) \stackrel{(cs,\langle -\rangle)}{\leftarrow} W(b^2 \mathbb{R}) \,. \end{displaymath} The remaining fiber integration is then that exhibiting the equivalence of simplicial differential forms \begin{displaymath} \int_{\Delta^\bullet} : \mathbf{\flat}_{dR} \mathbf{B}^3 U(1)_{diff,simp} \stackrel{\simeq}{\to} \mathbf{\flat}_{dR} \mathbf{B}^3 U(1)_{diff,chn} \end{displaymath} that is described in some detail at . \end{proof} \hypertarget{as_a_bundle_2gerbe}{}\paragraph*{{As a bundle 2-gerbe}}\label{as_a_bundle_2gerbe} We indicate (for the moment) the way the Chern-Simons 3-bundle is realized as a [[bundle 2-gerbe]] (for instance in \hyperlink{CJMS}{CJMS} and \hyperlink{WaldorfCS}{Waldorf CS}) . One first constructs the canonical [[bundle gerbe]] $\mathcal{G} \to G$ on the Lie group and notices that (more or less implicitly by recourse to its [[delooping]] 2-gerbe on $\mathbf{B}G$) that this has a \emph{multiplicative structure} . Using this we see that for $P \to X$ any $G$-[[principal bundle]] and $P^{[2]} : = P \times_X P \to P \times G$ the principality isomorphism, the pullback of $\mathcal{G}$ along \begin{displaymath} f : P^{[2]} \to P \times G \stackrel{p_2}{\to} G \end{displaymath} serves to provide the diagram \begin{displaymath} \itexarray{ f^* \mathcal{G} \\ \downarrow \\ P^{[2]} &\stackrel{\to}{\to}& P \\ && \downarrow \\ && X } \end{displaymath} on which the pullback of the multiplicative structure on $\mathcal{G}$ induces the structure of a bundle 2-gerbe, in that we get morphisms of bundle gerbes \begin{displaymath} \mu : \pi_0^{\ast} f^{\ast} \mathcal{G} \otimes \pi_2^{\ast} f^{\ast} \mathcal{G} \to \pi_2^* f^* \mathcal{G} \end{displaymath} that are associative up to a higher coherent morphisms, etc. \hypertarget{in_complex_analytic_geometry__for_complex_lie_groups}{}\subsubsection*{{In complex analytic geometry -- For complex Lie groups}}\label{in_complex_analytic_geometry__for_complex_lie_groups} (\ldots{}) (\hyperlink{Brylinski00}{Brylinski 00}) (\ldots{}) \hypertarget{ForReductiveAlgebraicGroups}{}\subsubsection*{{In arithmetic geometry -- For reductive groups}}\label{ForReductiveAlgebraicGroups} For [[reductive algebraic group]] $G$ there is no sensible element in $H^3(\mathbf{B}G, \mathbb{G}_m)$, but there is the following. Write $K_2(R)$ for the degree-2 [[algebraic K-theory]] [[group]] of a [[commutative ring]] (e.g. \href{algebraic+K-theory#Isely05}{Isely 05, section 4}) and write $\mathbf{K}_2$ for corresponding [[abelian sheaf]] on the suitable [[etale site]] (e.g. \hyperlink{DeligneBrylinski01}{Deligne-Brylinski 01, page 6}). Then \begin{displaymath} H^4(B G_{\mathbb{C}}, \mathbb{Z}) \simeq H^2(\mathbf{B}G, \mathbf{K}_2) \end{displaymath} This is (\hyperlink{HKLV98}{HKLV 98, theorem 4.11}) also (\hyperlink{DeligneBrylinski01}{Deligne-Brylinski 01}), going back to (\hyperlink{Bloch80}{Bloch 80}). See also (\hyperlink{Kapranov00}{Kapranov 00, (2.1)}, \href{http://mathoverflow.net/a/24862/381}{MO discussion}). Notice that over $\mathbb{C}$ the [[Beilinson regulator]] $c_{1,2}$ (e.g. \hyperlink{Brylinski94}{Brylinski 94, theorem, 3.3}) relates \begin{displaymath} H^1(X, \mathbf{K}_2) \longrightarrow H^2(X, \mathbb{G}_m) \,. \end{displaymath} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item The [[higher parallel transport]] of a Chern-Simons circle 3-bundle is the [[action functional]] for [[Chern-Simons theory]]. \item For the case that $G = O$ is the [[orthogonal group]] and $X \to \mathbf{B}O$ the classifying map of the [[tangent bundle]] of $X$, a trivialization of the corresponding Chern-Simons 3-bundle is a [[string structure]] on $X$. A trivialization of the Chern-Simons 3-bundle \emph{with connection} is a [[differential string structure]] on $X$. For products of torus groups and the [[cup product]] class the same construction yields the [[T-duality 2-group]] \item [[universal Chern-Simons n-bundle]] \begin{itemize}% \item \textbf{Chern-Simons circle 3-bundle} \item [[Chern-Simons circle 7-bundle]] \end{itemize} \end{itemize} The CS 3-bundle 3-connection is the [[extended Lagrangian]] for ordinary $G$-[[Chern-Simons theory]]. See there for more. \begin{itemize}% \item [[supergravity C-field]] \item [[Yang monopole]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{for_lie_groups}{}\subsubsection*{{For Lie groups}}\label{for_lie_groups} \hypertarget{the_cechdeligne_cohomology_realization}{}\paragraph*{{The Cech-Deligne cohomology realization}}\label{the_cechdeligne_cohomology_realization} As cocycles in [[Cech cohomology|Cech]]-[[Deligne cohomology]] the Chern-Simons 2-gerbe has been constructed explicitly in \begin{itemize}% \item [[Jean-Luc Brylinski]] and Dennis McLaughlin, \emph{A geometric construction of the first Pontryagin class} (1993) (\href{http://www.math.uni-hamburg.de/home/schreiber/Brylinski-McLaughlin-I.pdf}{pdf}) \end{itemize} as a special case of the general construction in \begin{itemize}% \item [[Jean-Luc Brylinski]] and Dennis McLaughlin, \emph{Cech cocycles for characteristic classes} , Communications in Mathematical Phiysics, Volume 178, Number 1, (\href{http://www.springerlink.com/content/762g1m76jp6425x5/}{Springer}) \end{itemize} \hypertarget{2GerbeReferences}{}\paragraph*{{The 2-gerbe realization}}\label{2GerbeReferences} Conceived as a genuine [[gerbe]] the Chern-Simons 2-gerbe appears in \begin{itemize}% \item [[Jean-Luc Brylinski]] and Dennis McLaughlin, \emph{The geometry of degree-4 characteristic classes and of line bundles on loop spaces II} (\href{http://www.math.uni-hamburg.de/home/schreiber/Brylinski-McLaughlin-Duke-II.pdf}{pdf}). \end{itemize} Among the first references to apply specifically [[bundle gerbe]] technology to this construction is \begin{itemize}% \item [[Alan Carey]], Stuart Johnson, [[Michael Murray]], [[Danny Stevenson]] and [[Bai-Ling Wang]], \emph{Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories} Communications in Mathematical Physics, 259 (3). (2005) (\href{}{arXiv}) \end{itemize} This was later refined in \begin{itemize}% \item [[Konrad Waldorf]], \emph{Multiplicative Bundle Gerbes with Connection} (\href{http://arxiv.org/abs/0804.4835}{arXiv:0804.4835}) \end{itemize} Here are some slides from talks: \begin{itemize}% \item [[Konrad Waldorf]], \emph{Multiplicative gerbes and Chern-Simons theory} (\href{http://www.konradwaldorf.de/docs/bonn.pdf}{pdf}) \end{itemize} \begin{itemize}% \item [[Krzysztof Gawedzki]], \emph{Wess-Zumino-Witten and Chern-Simons theories for non-simply connected Lie groups} (\href{http://dftuz.unizar.es/ftzar/activities/highenergy09_talks/gawedzki.pdf}{pdf}) \end{itemize} The full Chern-Simons circle 3-connection on the full [[moduli stack]] of $G$-[[principal connections]] $\mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$ was then constructed in \begin{itemize}% \item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech Cocycles for Differential characteristic Classes]]}, Advances in Theoretical and Mathematical Phyiscs, Volume 16 Issue 1 (2012) (\href{http://arxiv.org/abs/1011.4735}{arXiv:1011.4735}) \end{itemize} Exposition of this and further developments are in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:A higher stacky perspective on Chern-Simons theory]]} \end{itemize} \hypertarget{ReferencesForComplexReductiveGroups}{}\subsubsection*{{For complex reductive groups}}\label{ReferencesForComplexReductiveGroups} Discussion in [[complex analytic geometry]] of multiplicative Chern-Simons [[holomorphic line 2-bundles]] is in \begin{itemize}% \item [[Jean-Luc Brylinski]], around theorem 5.4.10 (p. 226-227) of \emph{Loop spaces and characteristic classes}, Birkh\"a{}user \item [[Jean-Luc Brylinski]], \emph{Gerbes on complex reductive Lie groups} (\href{http://arxiv.org/abs/math/0002158}{arXiv:math/0002158}) \end{itemize} The relevant [[Beilinson regulator]] is discussed also in \begin{itemize}% \item [[Jean-Luc Brylinski]], \emph{Holomorphic gerbes and the Beilinson regulator}, Ast\'e{}risque 226 (1994): 145-174 ([[Brylinski94.pdf:file]]) \end{itemize} \hypertarget{for_reductive_algebraic_groups}{}\subsubsection*{{For reductive algebraic groups}}\label{for_reductive_algebraic_groups} For the case of [[reductive algebraic groups]]: \begin{itemize}% \item [[Spencer Bloch]], \emph{The dilogarithm and extensions of Lie algebras}, Algebraic K-Theory Evanston 1980 Lecture Notes in Mathematics Volume 854, 1981, pp 1-23 (\href{http://link.springer.com/chapter/10.1007%2FBFb0089515}{publisher}) \item [[Hélène Esnault]], [[Bruno Kahn]], [[Marc Levine]], [[Eckart Viehweg]], \emph{The Arason invariant and mod 2 algebraic cycles}, J. Amer. Math. Soc. 11 (1998), 73-118 (\href{https://www.uni-due.de/~bm0032/publ/Arason.pdf}{pdf},\href{http://www.ams.org/journals/jams/1998-11-01/S0894-0347-98-00248-3/}{publisher page}) \item [[Mikhail Kapranov]], \emph{The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups} (\href{http://arxiv.org/abs/math/0001005}{arXiv.math/0001005}) \item [[Pierre Deligne]], [[Jean-Luc Brylinski]], \emph{Central extensions of reductive groups by $K_2$}, Publications Math\'e{}matiques de l'IH\'E{}S 2001 (\href{http://publications.ias.edu/node/424}{web}, \href{http://publications.ias.edu/sites/default/files/78_ReductiveGroupsK2.pdf}{pdf}) \item \href{http://mathoverflow.net/a/24862/381}{MO discussion} \end{itemize} [[!redirects Chern-Simons bundle 2-gerbe]] [[!redirects Chern-Simons circle 3-bundle with connection]] [[!redirects universal Chern-Simons circle 3-connection]] [[!redirects universal Chern-Simons 3-connection]] [[!redirects universal Chern-Simons circle 3-bundle with connection]] [[!redirects Chern-Simons line 3-bundle]] [[!redirects Chern-Simons line 3-bundles]] \end{document}