\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \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-Weil theory in Smooth∞Grpd} \begin{quote}% much of the material below has been or is being reworked into the entries [[Smooth∞Grpd]] and [[connection on a smooth principal ∞-bundle]] \end{quote} \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{Motivation}{Motivation}\dotfill \pageref*{Motivation} \linebreak \noindent\hyperlink{fractional_differential_classes}{Fractional differential classes}\dotfill \pageref*{fractional_differential_classes} \linebreak \noindent\hyperlink{higher_differential_spin_structures}{Higher differential spin structures}\dotfill \pageref*{higher_differential_spin_structures} \linebreak \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{PreparatoryConcepts}{Preparatory concepts}\dotfill \pageref*{PreparatoryConcepts} \linebreak \noindent\hyperlink{ChernWeil}{$\infty$-Chern-Weil theory}\dotfill \pageref*{ChernWeil} \linebreak \noindent\hyperlink{InfinityLieAlgebraConnection}{$\infty$-Lie algebra valued connections}\dotfill \pageref*{InfinityLieAlgebraConnection} \linebreak \noindent\hyperlink{InfChernWeil}{Curvature characteristics}\dotfill \pageref*{InfChernWeil} \linebreak \noindent\hyperlink{higher_order_chernsimons_forms}{Higher order Chern-Simons forms}\dotfill \pageref*{higher_order_chernsimons_forms} \linebreak \noindent\hyperlink{ChernCharacter}{Chern character}\dotfill \pageref*{ChernCharacter} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{principal_1bundles_with_connection}{Principal 1-bundles with connection}\dotfill \pageref*{principal_1bundles_with_connection} \linebreak \noindent\hyperlink{principal_2bundles_with_connection}{Principal 2-bundles with connection}\dotfill \pageref*{principal_2bundles_with_connection} \linebreak \noindent\hyperlink{DiffStringStruc}{Twisted differential $String-$ and $Fivebrane$-structures}\dotfill \pageref*{DiffStringStruc} \linebreak \noindent\hyperlink{StringStructure}{The string-lifting Chern--Simons $3$-bundle with connection}\dotfill \pageref*{StringStructure} \linebreak \noindent\hyperlink{differential_string_structures}{Differential string structures}\dotfill \pageref*{differential_string_structures} \linebreak \noindent\hyperlink{FivebraneStructure}{The Fivebrane-lifting Chern-Simons 7-bundle with connection}\dotfill \pageref*{FivebraneStructure} \linebreak \noindent\hyperlink{DiffFivebraneStrucs}{Differential fivebrane structures}\dotfill \pageref*{DiffFivebraneStrucs} \linebreak \noindent\hyperlink{chernsimons_theory}{$\infty$-Chern-Simons theory}\dotfill \pageref*{chernsimons_theory} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak In every [[cohesive (∞,1)-topos]] there is an . We discuss the concrete realization of this in the cohesive $(\infty,1)$-topos [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s. This s the case that subsumes ordinary [[Chern-Weil theory]] of [[SmoothMfd|smooth]] [[principal bundle]]s with [[connection on a bundle|connection]] and generalizes it to [[connections on smooth principal ∞-bundles]]. \hypertarget{Motivation}{}\subsection*{{Motivation}}\label{Motivation} The central motivation for the study of a higher generalization of ordinary [[Chern-Weil theory]] is the interest in extending the [[Chern-Weil homomorphism]] for a given [[Lie group]] $G$ to the higher connected covers of $G$ through the whole [[Whitehead tower]] of $G$. Beyond the simply connected cover, these higher connected covers are still [[topological group]]s but fail to be (finite dimensional) [[Lie group]]s. They do however have natural realizations as [[smooth ∞-group]]s. Higher Chern-Weil theory is the extension of [[Chern-Weil theory]] from Lie groups to such smooth $\infty$-groups. It allows the refinement of differential characteristic classes to \emph{fractional} differential characteristic classes, that capture finer cohomological information. \hypertarget{fractional_differential_classes}{}\subsubsection*{{Fractional differential classes}}\label{fractional_differential_classes} We give some examples of such \emph{fractional characteristic classes} that occur in practice. It is a familiar classical fact that the first [[Pontryagin class]] \begin{displaymath} p_1 : \mathcal{B}SO \to \mathcal{B}^4 \mathbb{Z} \,, \end{displaymath} which represents the generator of the fourth [[integral cohomology]] of the [[classifying space]] $\mathcal{B} SO$ of the [[special orthogonal group]] allows a division by 2 when pulled back one step along the [[Whitehead tower]] to the classifying space of the [[spin group]], in that there is a [[commuting diagram]] \begin{displaymath} \itexarray{ \mathcal{B} Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathcal{B}^4 \mathbb{Z} \\ \downarrow && \downarrow^{\mathrlap{\cdot 2}} \\ \mathcal{B} SO &\stackrel{p_1}{\to}& \mathcal{B}^4 \mathbb{Z} } \,, \end{displaymath} in [[Top]], where the top horizontal morphism represents a generator of the 4th integral cohomology of the classifying space of the [[spin group]] and the right vertical morphism is induced by multiplication by 2 on the additive group of [[integer]]s. This means that for $X$ [[manifold]] with [[spin structure]] exhibited by a classifying map $\hat g$ \begin{displaymath} \itexarray{ && \mathcal{B} Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathcal{B}^4 \mathbb{Z} \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow && \downarrow^{\mathrlap{\cdot 2}} \\ X &\stackrel{g}{\to}& \mathcal{B} SO &\stackrel{p_1}{\to}& \mathcal{B}^4\mathbb{Z} } \end{displaymath} of its [[tangent bundle]] $T X$, the characteristic class $p_1(T X) : X \stackrel{g}{\to} \mathcal{B}SO \stackrel{p_1}{\to} \mathcal{B}^4 \mathbb{Z}$ of $T X$ regarded as an $SO$-[[associated bundle]] contains less information than the class $\frac{1}{2}p_1(T X) : X \stackrel{\hat g}{\to} \mathcal{B}Spin \stackrel{\frac{1}{2}p_1}{\to} \mathcal{B}^4 \mathbb{Z}$. For instance if the 4th cohomology of $X$ happens to be 2-[[torsion]], the former class entirely vanishes, while the latter need not. This familiar situation poses no problem to classical [[Chern-Weil theory]], because both the [[special orthogonal group]] as well as the [[spin group]] of course have canonical structures of [[Lie group]]s, so that the [[Chern-Weil homomorphism]] may be applied to either. We shall write $\mathbf{B} \mathrm{Spin}$ for the smooth refinement of the classifying space $B \mathrm{Spin}$: the delooping Lie groupoid of $\mathrm{Spin}$ or equivalently the moduli stack for smooth $\mathrm{Spin}$-principal bundles. Here and in the following the boldface indicates smooth (or otherwise cohesive) refinements. Accordingly, there is a smooth refinement $\frac{1}{2}\mathbf{p} : \mathbf{B} \mathrm{Spin} \to \mathbf{B}^3 U(1)$ of the first Pontryagin class, which takes smooth $\mathrm{Spin}$-principal bundles to their first Pontryagin class. This in turn has has a further differential refinement $\frac{1}{2}{\hat {\mathbf{p}}} : \mathbf{B}\mathrm{Spin}_{\mathrm{conn}} \to \mathbf{B}^3 U(1)_{\mathrm{conn}}$ that takes $\mathrm{Spin}$-principal bundles with connection to their Chern-Simons 2-gerbes with connection. All this is still captured by the traditional (refined) Chern-Weil homomorphism. But this is no longer the case as we keep climbing up the [[Whitehead tower]] of the [[orthogonal group]]. In the next step the second [[Pontryagin class]] $p_2 : \mathcal{B}SO \to \mathcal{B}^8 \mathbb{Z}$ may be divided by 6 when pulled back to the classifying space of the [[string group]] () \begin{displaymath} \itexarray{ \mathcal{B} String &\stackrel{\frac{1}{6}p_2}{\to}& \mathcal{B}^8 \mathbb{Z} \\ \downarrow && \downarrow^{\mathrlap{\cdot 6}} \\ \mathcal{B} SO &\stackrel{p_2}{\to}& \mathcal{B}^8 \mathbb{Z} } \,, \end{displaymath} As before, this means that if a space $X$ admits a [[string structure]], then the characteristic class $p_2(X)$ contains less information than the fractional refinement $\frac{1}{6}p_2(X)$ that it admits. In particular, the former may vanish if the degree 8 cohomology group of $X$ has 6-[[torsion]], while the latter need not vanish. For purposes of ordinary cohomology this is no problem, but for the differential refinement by ordinary [[Chern-Weil theory]] it is: the [[string group]] does not admit a [[Lie group]] structure that would make it a smooth version of the homotopy fiber of $\frac{1}{2}p_1$ and hence standard [[Chern-Weil theory]] cannot produce the differential refinement of the fractional class $\frac{1}{6}p_2$. But $\infty$-Chern-Weil theory can: there is a natural smooth refinement of the [[string group]] to a [[Lie 2-group]]: the [[string 2-group]]. We write $\mathbf{B}String$ for the corresponding [[delooping]] [[∞-Lie groupoid]]. The fractional second Pontryagin class does lift to this smooth refinement to produce a [[characteristic class]] \begin{displaymath} \frac{1}{6}\mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1) \end{displaymath} internal to $\mathbf{H} =$ [[?LieGrpd]]. Since this now lives in a smooth context, it does now have a differential Chern-Weil refinement \begin{displaymath} \frac{1}{6}\hat \mathbf{p}_2 : \mathbf{H}_{conn}(-, \mathbf{B}String) \to \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1)) \end{displaymath} that takes smooth $String$-[[principal 2-bundle]]s with 2-connection to degree 8-cocycles in [[ordinary differential cohomology]]. This kind of refinement we discuss in a bit more detail in the next section. \hypertarget{higher_differential_spin_structures}{}\subsubsection*{{Higher differential spin structures}}\label{higher_differential_spin_structures} These refined differential invariants of fractional characteristic classes are relevant in the discussion of higher differential spin structures. (See the first part of ( for a review).) Ordinary [[spin structure]]s on a [[manifold]] may be understood as trivializations of what are called [[quantum anomaly]] [[Pfaffian line bundle]]s on the configuration space of the spinning [[relativistic particle|quantum particle]] propagating on that manifold. (This physical origin is after all the origin of the term \emph{spin structure} .) When these point-like super-particles are generalized to higher-dimensional $p$-[[brane]]s, the trivialization of the corresponding Pfaffian line bundles correspond to [[string structure]]s for $p = 1$ (this goes back to () and () and has been made rigorous in () then to [[fivebrane structure]]s for $p = 5$ ()). More precisely, the Pfaffian line bundles appearing here come equipped with a [[connection on a bundle|connection]], and what matters is a trivialization of these bundles as bundles with connection. This refinement translates to differential refinements of the string structures and the fivebrane structures on $X$. The differential form data of a twisted [[differential string structure]] constitutes what in the physics literature is called the [[Green-Schwarz mechanism]]. While this still can and has been captured with tools of ordinary Chern-Weil theory and ordinary differential cohomology (, ) it has a natural formulation in higher Chern-Weil theory. Going beyond that, the [[nLab:dual heterotic string theory|magnetic dual]] Green-Schwarz mechanism can be seen to encode a twisted differential fivebrane structure and this is not practical to be studied without some higher geometry. The following restates this in a bit more technical detail. For $G = Spin$ the [[spin group]], the first nontrivial [[characteristic class]] is the first fractional [[Pontryagin class]] given by a [[cocycle]] $\frac{1}{2}p_1 : \mathcal{B}G \to K(\mathbb{Z}, 4)$ in ordinary [[integral cohomology]] $H^4(\mathcal{B}Spin, \mathbb{Z})$. This induces a map \begin{displaymath} H^1(X, Spin) = H(X, \mathcal{B}Spin) \to H^4(X, \mathbb{Z}) \end{displaymath} from isomorphism classes of topological $Spin$-[[principal bundle]]s to degree 4 integral cohomology. If we assume that $X$ is a [[smooth manifold]] then we may consider the set \begin{displaymath} Spin Bund(X)/ \sim = H(X,\mathbf{B}Spin) \end{displaymath} of [[isomorphism]]-classes of \emph{smooth} $Spin$-[[principal bundle]]s. Here and in all of the following, the boldface in ``$\mathbf{B}G$'' indicates a refinement, here of the bare classifying space $\mathcal{B}G$ to a smooth incarnation. Then ordinary [[Chern-Weil theory]] provides a refinement of the fractional Pontryagin class $H(X, \mathbf{B}Spin) \to H^4(X,\mathbb{Z})$ to a map to [[ordinary differential cohomology]] $H_{diff}^4(X)$ \begin{displaymath} \frac{1}{2} \hat p_1 : H(X, \mathbf{B}Spin) \to H_{diff}^4(X) \,. \end{displaymath} The first point of passing to a [[higher category theory]]-refinement of this situation is that it allows to refine, in turn, these morphisms of cohomology \emph{sets} to morphisms \begin{displaymath} \frac{1}{2} \mathbf{p}_1 : \mathbf{H}(X, \mathbf{B}Spin) \to \mathbf{H}(X,\mathbf{B}^3 U(1)) \end{displaymath} and \begin{displaymath} \frac{1}{2} \hat \mathbf{p}_1 : \mathbf{H}_{conn}(X, \mathbf{B}Spin) \to \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1)) \end{displaymath} of [[cocycle]] [[∞-groupoid]]s: here $\mathbf{H}(X,\mathbf{B}G)$ is the [[groupoid]] whose objects are smooth $Spin$-[[principal bundle]]s, and whose morphisms are smooth homomorphisms between these. Similarly $\mathbf{H}(X,\mathbf{B}^3 U(1))$ denotes the [[3-groupoid] whose objects are smooth -[[principal ∞-bundles|principal 3-bundles]], while $\mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))$ is accordingly the [[3-groupoid]] whose objects are [[circle n-bundle with connection|circle 3-bundles with connection]], whose morphisms are homomorphisms between these, whose 2-morphisms are higher homotopies between those. The original morphism of cohomology sets is the [[decategorification]] of this, the restriction to connected components: \begin{displaymath} \frac{1}{2}p_1 = \pi_0(\frac{1}{2}\mathbf{p}_1) \,. \end{displaymath} This refinement to cocylce $\infty$-groupoids notably has the consequence that it allows us to produce the [[homotopy fiber]]s of these morphisms. To see the relevance of this, recall (from \emph{[[string structure]]} ) that the [[homotopy fiber]] of the bare fractional Pontryagin class, which is the [[(∞,1)-pullback]]/[[homotopy pullback]] \begin{displaymath} \itexarray{ \mathbf{H}(X,\mathbf{B}String) &\to& * \\ \downarrow &\swArrow_\simeq& \downarrow \\ \mathbf{H}(X,\mathbf{B}G) &\stackrel{\frac{1}{2} \mathbf{p}_1}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) } \,, \end{displaymath} defines the $\infty$-groupoid $\mathbf{H}(X, \mathbf{B}String)$ of [[string structure]]s on $X$ ( \emph{smooth} , but not \emph{differential} ). We can now replace the class $\frac{1}{2}\mathbf{p}_1$ by its differential refinement $\frac{1}{2}\hat \mathbf{p}_1$ and obtain an [[∞-groupoid]] $String_{diff}(X)$ that differentially refines the 2-groupoid $\mathbf{H}(X,\mathbf{B}String)$ of String-structures as the [[(∞,1)-pullback]] \begin{displaymath} \itexarray{ String_{diff}(X) &\to& * \\ \downarrow &\swArrow_\simeq& \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}G) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) } \,. \end{displaymath} This $String_{diff}(X)$ we may call the $\infty$-groupoid of \emph{[[differential string-structures]]} . A cocycle in there is naturally identified with a tuple consisting of \begin{itemize}% \item a smooth $Spin$-[[principal bundle]] $P \to X$ with [[connection on a bundle|connection]] $\nabla$; \item the [[Chern-Simons 2-gerbe]] with connection $CS(\nabla)$ induced by this; \item a choice of trivialization of this Chern-Simons 2-gerbe -- this is the [[homotopy]] [[2-morphism]] in the middle of the above pullback diagram. \end{itemize} We may think of this as a refinement of [[secondary characteristic class]]es: the first Pontryagin [[curvature characteristic form]] $\langle F_\nabla \wedge F_\nabla \rangle$ itself is constrained to vanish, and so the [[Chern-Simons form]] 3-connection itself constitutes cohomological data. So far this uses mostly just a little bit of [[(∞,1)-category theory]] or at least some [[homotopy theory]]. The first glimpse of something beyond ordinary [[Chern-Weil theory]] appearing is the $\infty$-groupoid $\mathbf{H}(X,\mathbf{B}String)$ which may be thought of as the [[2-groupoid]] of \emph{smooth} [[string 2-group]]-[[principal 2-bundle]]s. But suppose we fix an $X$ such that $H(X, \mathbf{B}String)$ is nontrivial. Then we can continue the proceed to higher degrees: the next topological characteristic class is the second fractional [[Pontryagin class]] $\frac{1}{6}p_2 : \mathcal{B}String \to \mathcal{B}^7 U(1)$. Since the [[string group]] does not have the structure of a [[Lie group]], this cannot be refined to [[differential cohomology]] using ordinary [[Chern-Weil theory]]. However, in terms of $\infty$-Chern-Weil theory it can: we may obtain a differential refinement \begin{displaymath} \frac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn} \end{displaymath} that maps smooth [[string 2-group]]-[[principal 2-bundle]]s with 2-connectins to their \hyperlink{FivebraneStructure}{Chern-Simons circle 7-bundle with connection}. This is an example of the higher version of the [[Chern-Weil homomorphism]]. And naturally we are then entitled to form its [[homotopy fiber]]s and produce the [[n-groupoid|7-groupoid]] of \emph{\hyperlink{DiffFivebraneStrucs}{differential fivebrane structures}} -- $Fivebrane_{diff}(X)$. For that notice (see [[fivebrane structure]]) that the homotopy fiber of the smooth but non-differential cocycles \begin{displaymath} \itexarray{ \mathbf{H}(X, \mathbf{B}Fivebrane) &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{H}(X, \mathbf{B}String) &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{H}(X, \mathbf{B}^7 U(1)) } \end{displaymath} is the [[n-groupoid|7-groupoid]] of smooth [[fivebrane structure]]s on $X$. Its differential refinement \begin{displaymath} \itexarray{ Fivebrane_{diff}(X) &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{H}_{conn}(X, \mathbf{B}String) &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^7 U(1)) } \end{displaymath} we may therefore call the 7-groupoid $Fivebrane_{diff}(X)$ of \emph{differential fivebrane structures} . Cocycles in here are naturally identified with tuples of \begin{itemize}% \item a $String$-[[principal 2-bundle]] $P \to X$, equipped with a \hyperlink{InfinityLieAlgebraConnection}{2-connection} $\nabla$; \item the Chern-Simons [[circle n-bundle with connection|circle 7-bundle]] $CS_7(\nabla)$ with connection induced by it; \item a choice of trivialization of $CS_7(\nabla)$. \end{itemize} These are the kind of structures that $\infty$-Chern-Weil theory studies. \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} Ordinary [[Chern-Weil theory]] is about refinements of [[characteristic class]]es of $G$-[[principal bundle]]s for $G$ a [[Lie group]] (equivalently of the [[classifying space]] $\mathcal{B}G$ of that Lie group) from ordinary [[cohomology]] to [[differential cohomology]]. Under \emph{$\infty$-Chern-Weil theory} we want to understand the generalization of this to [[(∞,1)-category theory]]: where [[Lie group]]s are generalized to [[∞-Lie group]]s, [[Lie algebra]]s are generalized to [[∞-Lie algebra]]s and [[principal bundle]]s to [[principal ∞-bundle]]s. So $\infty$-Chern-Weil theory produces [[differential cohomology]]-refinements of [[characteristic class]]es of $G$-[[principal ∞-bundle]]s for $G$ an [[∞-Lie group]], equivalently of the corresponding [[classifying space]]s $\mathcal{B}G$. Ordinary Chern-Weil theory is formulated in the context of [[differential geometry]]. We need to widen this context somewhat in order that it can accomodate the relevant higher structures and so we place ourselves in the context of the [[(∞,1)-topos]] $\mathbf{H} =$ [[?LieGrpd]] of [[∞-Lie groupoids]]. In every $(\infty,1)$-topos that admits a notion of differential cohomology, there is a general abstract notion of refinement of [[characteristic class]]es in [[cohomology]] to [[curvature characteristic forms|curvature characteristic classes]] in [[ordinary differential cohomology]]. The main construction in ∞-Chern-Weil theory is a concrete \emph{model} or \emph{presentation} of this abstract operation. This model is constructed in terms of [[Lie integration]] of objects in [[∞-Lie algebra cohomology]]. This construction is the higher analog of the [[Chern-Weil homomorphism]]. Its crucial intermediate step is the definition and construction of \emph{[[connection on an ∞-bundle|∞-connections]]} on [[principal ∞-bundle]]s. This model itself is after all built on concrete familiar constructions in [[differential geometry]] and can be studied and appreciated in itself without recourse to the higher topos theory that we claim it provides a model for. The so inclined reader can ignore all the general abstract discussion in the following and concentrate on the concrete differential geometry. ere is how this entry here proceeds. A warmup for the full theory that connects to classical constructions is given at \begin{itemize}% \item \hyperlink{PreparatoryConcepts}{Introduction} \end{itemize} Then in \begin{itemize}% \item \hyperlink{ChernWeil}{∞-Chern-Weil theory} \end{itemize} we discuss the general definition of $\infty$-connections and of the Chern-Weil homomorphism and discuss some general properties. Then we turn to discussing \begin{itemize}% \item \hyperlink{Examples}{Examples} \end{itemize} \hypertarget{PreparatoryConcepts}{}\subsection*{{Preparatory concepts}}\label{PreparatoryConcepts} General $\infty$-Chern-Weil theory, as described below, is naturally formulated in the context of [[(infinity,1)-topos]]-theory and some of its aspects can only be understood from that perspective. However, unwinding the abstract higher topos theoretic concepts in terms of 1-categoriecal models yields concrete structures in familiar contexts of [[differential geometry]] that connect to various classical and familiar concepts. Since a full appreciation of the abstract formulation benefits from having a feeling for how these concrete models work out, the reader may at this point wish to look into some such basic aspects. These may be found behind the following link \begin{itemize}% \item [[infinity-Chern-Weil theory -- preparatory concepts]]. \end{itemize} \hypertarget{ChernWeil}{}\subsection*{{$\infty$-Chern-Weil theory}}\label{ChernWeil} For $G,A$ [[nLab:∞-group]]s in an [[∞-connected (∞,1)-topos]] $\mathbf{H}$ with [[nLab:delooping]]s $\mathbf{B}G$ and $\mathbf{B}A$, respectively, every [[nLab:characteristic class]] $c : \mathbf{B}G \to A$ serves to pull back the $curv_A : A \to \mathbf{\flat}_{dR} \mathbf{B}A$ to an $curv_A\circ c : \mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}A$ on $\mathbf{B}G$. For $G$ an ordinary [[nLab:Lie group]] regarded naturally as an object in $\mathbf{H} =$ [[nLab:?LieGrpd]], we show that the ordinary [[nLab:Chern-Weil homomorphism]] for $G$-[[nLab:principal bundle]]s may be understood as a concrete \emph{model} for this simple abstract situation, which applies to those characteristic classes $c$ that happen to be in the image of the . More generally, this construction applies for $G$ an [[nLab:∞-Lie group]] with [[nLab:∞-Lie algebra]] $\mathfrak{g}$ and $c$ a characteristic class on $\mathbf{B}G$ that arises from Lie integration of a cocycle in the [[nLab:∞-Lie algebra cohomology]] of $\mathfrak{g}$. The ordinary [[nLab:Chern-Weil homomorphism]] uses a [[nLab:connection on a bundle]] $\nabla$ as an intermediate tool for interpolating from a $G$-[[nLab:principal bundle]] to its curvature characteristic, represented by the [[nLab:curvature characteristic form]] $\langle F_\nabla \rangle$, where $F_\nabla$ is the [[nLab:curvature]] of $\nabla$ and $\langle - \rangle$ is an [[nLab:invariant polynomial]] on $\mathfrak{g}$. The choice of connection in this construction may be understood as providing a correspondence space in the following construction. We know from the discussion of abelian differential cohomology \hyperlink{AbGerbe}{above} that the intrinsic morphism \begin{displaymath} \mathbf{B}^n \mathbb{R}/\mathbb{Z} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}/\mathbb{Z} \end{displaymath} in $\mathbf{H} = \infty LieGrpd$ is modeled in $[CartSp^{op}, sSet]_{proj,cov}$ by the correspondence \begin{displaymath} \itexarray{ \mathbf{B}^n \mathbb{R}/\Gamma_{diff,simp} &\stackrel{curv}{\to}& \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}_{simp} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n \mathbb{R}/\Gamma_{simp} } \,. \end{displaymath} If we write \begin{displaymath} \exp(b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R})) : \mathbf{cosk}_{n+1}( (U,[k]) \mapsto \left\{ \itexarray{ C^\infty(U)\otimes \Omega^\bullet(\Delta^k) &\leftarrow& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(b^{n-1}\mathbb{R}) } \right\} ) \end{displaymath} and so forth, then this correspondence is \begin{displaymath} \itexarray{ \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(b^{n-1}\mathbb{R}\to *) } \,. \end{displaymath} If now $\mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\mathbb{Z}$ is modeled by the Lie integration of a cocycle $\mu$ on a Lie $k$-algebra \begin{displaymath} \mathbf{cosk}_{k+1} \exp(\mathfrak{g}) \stackrel{\exp(\mu)}{\to} \exp(b^{n-1}\mathbb{R})/\Gamma = \exp(b^{n-1}\mathbb{R} \to *)/\Gamma \end{displaymath} for $k \geq n-1$, then the total intrinsic differential form $\mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\Gamma \to \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}$ is modeled by the zig-zag of morphisms \begin{displaymath} \itexarray{ && \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{cosk}_{k+1}\exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to *) } \end{displaymath} in $[CartSp^{op}, sSet]$. In order to compute with such zig-zags of morphisms, in particular in order to compute [[nLab:homotopy fiber]]s, it is helpful to complete this to a single correspondence. There is a fairly evident choice for the tip of this total corresponence, namely \begin{displaymath} \mathbf{B}G_{diff} := \mathbf{cosk}_{n+1} \exp(\mathfrak{g} \to inn(\mathfrak{g})) \,. \end{displaymath} It remains to complete the square and extend the [[nLab:∞-Lie algebra cohomology|∞-Lie algebra cocycle]] $\mu : \mathfrak{g} \to b^{n-1}\mathb{R}$ to a morphism $(\mathfrak{g} \to inn(\mathfrak{g})) \to (b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R}))$. This is accomplished by an [[nLab:invariant polynomial]] \begin{displaymath} \langle -\rangle_\mu : inn(\mathfrak{g}) \stackrel{(\langle - \rangle_\mu, cs_\mu)}{\to} inn(b^{n-1}\mathbb{R}) \to b^n \mathbb{R} \end{displaymath} which is in transgression with $\mu$, witnessed by the [[nLab:Chern-Simons form|Chern-Simons element]] $cs_\mu$. Using this, we obtain the total diagram \begin{displaymath} \itexarray{ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to inn(\mathfrak{g})) &\stackrel{(\langle - \rangle_\mu, cs_\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{cosk}_{k+1}\exp(\mathrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to *) } \,. \end{displaymath} By the fact that this commutes, we have that the correspondence \begin{displaymath} \left( \itexarray{ \mathbf{B}G_{diff,simp} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{simp} \\ \downarrow \\ \mathbf{B}G_{simp} } \right) \;\; := \;\; \left( \itexarray{ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to inn(\mathfrak{g})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow \\ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to *) } \right) \end{displaymath} in $[CartSp^{op}, sSet]_{proj,cov}$ models the intrinsic curvature characteristic form $\mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}$. We may identify cocycles with values in $\mathbf{B}G_{diff}$ as \emph{(pseudo)-$\infty$-connections} on the underlying $\mathbf{B}G$-cocycle. If their curvature is represented by a cocycle in $\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}$ which is given by a globally defined form, then these are \emph{genuine} $\infty$-connections. In either case, they serve as an intermediate step in computing the curvature characteristics. \hypertarget{InfinityLieAlgebraConnection}{}\subsubsection*{{$\infty$-Lie algebra valued connections}}\label{InfinityLieAlgebraConnection} \begin{quote}% The content of this section is at [[connection on an infinity-bundle]]. \end{quote} \hypertarget{InfChernWeil}{}\subsubsection*{{Curvature characteristics}}\label{InfChernWeil} \begin{udef} \textbf{(Chern-Weil curvature characteristics)} Let $\langle -\rangle : inn(\mathfrak{g}) \to b^{p} \mathbb{R}$ be an [[nLab:invariant polynomial]] on the [[nLab:∞-Lie algebra|Lie n-algebra]] $\mathfrak{g}$. Postcomposition with the corresponding diagram of dg-algebras \begin{displaymath} \itexarray{ CE(\mathfrak{g}) &\leftarrow& 0 \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& CE(b^p \mathbb{R}) } \end{displaymath} induces a morphism of [[nLab:simplicial presheaves]] \begin{displaymath} \langle F_{(-)} \rangle : \mathbf{B}G_{diff} \to \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1}\mathbb{R}_{simp} \end{displaymath} into the $(n+1)$-[[nLab:simplicial skeleton|coskeleton]] of the model for the de Rham coefficient object $\mathbf{\flat}_{dR}\mathbf{B}^{p+1}\mathbb{R}$ discussed \hyperlink{U1FromLieIntegration}{above}. For $\nabla : \hat X \to \mathbf{B}G_{diff}$ a connection, we call the induced intrinsic de Rham cocycle \begin{displaymath} \langle F_\nabla \rangle : \hat X \stackrel{\nabla}{\to} \mathbf{B}G_{diff} \stackrel{\langle -\rangle}{\to} \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1}\mathbb{R}_{simp} \end{displaymath} the Chern-Weil [[nLab:curvature characteristic form]] of $\nabla$ with respect to $\langle -\rangle$. \end{udef} \begin{ulemma} For $\nabla : \hat X \to \mathbf{B}G_{conn} \hookrightarrow \mathbf{B}G_{diff}$ a genuine connection, the induced curvature characteristic forms are globally defined closed forms, in that their cocycle factors through the sheaf $\Omega^{p+1}_{cl}(-)$ of closed $(p+1)$-forms: \begin{displaymath} \itexarray{ && \mathbf{B}G_{conn} &\stackrel{\langle F_{(-)}\rangle}{\to}& \Omega^{p+1}_{cl}(-) \\ & \nearrow & \downarrow && \downarrow \\ \hat X &\stackrel{}{\to}& \mathbf{B}G_{diff} &\stackrel{\langle F_{(-)}\rangle}{\to}& \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1} \mathbb{R}_{simp} } \,. \end{displaymath} \end{ulemma} \begin{proof} for given $(U,[k])$ notice that $\langle F_{\nabla}\rangle(U,[k]) \in \Omega^\bullet(U\times \Delta^k)$ is closed and for $\nabla$ a genuine connection has no leg along $\Delta^k$: for $\partial_t$ a vector field along $\Delta^k$ we have $\iota_{\partial_t} \langle F_A\rangle = 0$. Therefore the [[nLab:Lie derivative]] along a vector $\partial_t$ along the simplex vanishes: \begin{displaymath} \mathcal{L}_t \langle F_A\rangle = d \iota_t \langle F_A\rangle + \iota_t d \langle F_A\rangle = 0 \,. \end{displaymath} \end{proof} \begin{uremark} As for the groupal case \hyperlink{AbGerbesConnection}{above}, we hence find that the genuine $\infty$-connections are selected among all pseudo-connections as those whose curvature characteristic has a 0-[[nLab:truncated]] cocycle representative. \end{uremark} So a genuine $\infty$-Lie algebra valued connection is a cocycle with values in the $(n+1)$-coskeleton of the simplicial presheaf of diagrams, which over $U,[k]$ assigns the set of diagrams \begin{displaymath} \itexarray{ C^\infty(U \times \Delta k)_{vert} &\leftarrow& CE(\mathfrak{g}) &&& cocycle\;for\;underlying\;G-principal\;\infty-bundle \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(X) &\leftarrow& CE(inn(\mathfrak{g})) = W(\mathfrak{g}) &&& connection\;and\;curvature \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\leftarrow& inv(\mathfrak{g}) &&& curvature\;characteristic\;forms } \,, \end{displaymath} (with $\Omega^\bullet(U \times \Delta^k)_{vert}$ the dg-algebra of [[vertical differential form]]s on the bundle $U \times \Delta^k \to U$), where the top morphism encodes the cocycle for the underlying $G = \tau_n\exp(\mathfrak{g})$-[[nLab:principal ∞-bundle]], where the middle morphism encodes the connection data and the bottom morphism the [[nLab:curvature characteristic form]]s. Such $\infty$-Lie algebra valued connections were introduced in and further studied in . \hypertarget{higher_order_chernsimons_forms}{}\subsubsection*{{Higher order Chern-Simons forms}}\label{higher_order_chernsimons_forms} See at [[Chern-Simons form]] the section . \hypertarget{ChernCharacter}{}\subsubsection*{{Chern character}}\label{ChernCharacter} Above we have considered \hyperlink{InfinityLieAlgebraConnection}{∞-Lie algebra valued connections} and their \hyperlink{InfChernWeil}{curvature characteristic forms}. We now wish to show how these model the intrinsic [[schreiber:Chern character in an (∞,1)-topos]]. \begin{displaymath} ch_{\mathbf{B}G} : \mathbf{B}G \to \mathbf{\Pi}(\mathbf{B}G) \to \mathbf{\Pi}(\mathbf{B}G)\otimes R \,. \end{displaymath} Since our ambient [[(∞,1)-topos]] is assumed to be [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] we have in addition to the notion of [[Postnikov tower in an (∞,1)-category]] the notion of [[nLab:Whitehead tower in an (∞,1)-topos]]. Both notions are dual to each other: for $A \in \mathbf{H}$ any object and \begin{displaymath} \mathbf{\Pi}(A) \to \cdots \to \tau_{\leq 2}\mathbf{\Pi}(A) \to \tau_{\leq 1}\mathbf{\Pi}(A) \to \tau_{\leq 0}\mathbf{\Pi}(A) = * \end{displaymath} the [[nLab:Postnikov tower in an (∞,1)-category|intrinsic Postnikov tower]] of its [[path ∞-groupoid]], the [[nLab:pasting]] composite of [[nLab:(∞,1)-pullback]]s \begin{displaymath} \itexarray{ \vdots && && && \vdots \\ \downarrow && && && \downarrow \\ A_2 && &\to& \cdots &\to& \mathbf{B}\mathbf{\pi}_3(A) &\to& * \\ \downarrow && && && && \downarrow \\ A_1 && &\to& \cdots && && \mathbf{B}\mathbf{\pi}_2(A) &\to& * \\ \downarrow && && && && \downarrow && \downarrow \\ A &\to& \mathbf{\Pi}A &\to& \cdots &\to& \tau_{\leq 3} \mathbf{\Pi}A &\to& \tau_{\leq 2} \mathbf{\Pi}A &\to& \tau_{\leq 1} \mathbf{\Pi}A } \,, \end{displaymath} defines the [[nLab:Whitehead tower in an (∞,1)-topos|Whitehead tower]] \begin{displaymath} * \to \cdots \to A_3 \to A_2 \to A_1 \to A_0 = A \end{displaymath} of $A$. Since our $\mathbf{H}$ is assumed to be even [[nLab:∞-connected (∞,1)-topos|∞-connected]], the Postnikov tower of $\mathbf{\Pi}(A)$ is the image under $LConst : \infty Grpd \to \mathbf{H}$ of the ordinary [[nLab:Postnikov tower]] of $\Pi(A)$ in $\infty Grpd$. Accordingly, we have $\mathbf{B} \mathbf{\pi}_n(A) = LConst B^n \pi_n \Pi(A)$ The point now is that in $\mathbf{H} =$ [[nLab:?LieGrpd]] we may form smooh refinements of these discrete extensions: every discrete $(n+1)$-group $\mathbf{B}^{n+1}\mathbb{Z}$ we want to refine to a smooth $n$-group $\mathbf{B}^n U(1)$. By the discussion at , both have equivalent underlying $\infty$-groupoids \begin{displaymath} \Pi(\mathbf{B}^{n+1}\mathbb{Z}) \simeq \Pi(\mathbf{B}^n U(1)) \simeq K(\mathbb{Z},n+1) \,. \end{displaymath} For every direct summand abelian group $\mathbb{Z}$ in one of the $\mathbf{\pi}_n(A)$ we can ask for a refinement of the cocycle from coefficients $\mathbf{B}^n \mathbb{Z}$ to $\mathbf{B}^{n-1}\mathbb{R}/\mathbb{Z}$. This does not change the geometric realization, up to equivalence, but does change the smooth structure. And it allows to refine to differential coefficients by postcomposing further with $curv : \mathbf{B}^{n-1}\mathbb{R}/\mathbb{Z} \to \mathbf{\flat}_{dR}\mathbf{B}^{n}\mathbb{R}$. For instance for $A = \mathbf{B}Spin \in \mathbf{H} = \infty Lie Grpd$ the [[nLab:delooping]] of the [[nLab:spin group]], we refine the internal Whitehead tower to \begin{displaymath} \itexarray{ \vdots \\ \mathbf{B}Fivebrane &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}String &\to& \mathbf{B}^7 U(1) &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}Spin &\to& \cdots &\to& \mathbf{B}^3 U(1) } \,, \end{displaymath} where the deloopings of the [[nLab:string 2-group]] and the [[nLab:fivebrane 6-group]] appear. The result of such a smooth refinement is that we may apply the intrinsic curvature classes and the intrinsic de Rham theorem to obtain cocycles in realified cohomology, for instance \begin{displaymath} \mathbf{B}Spin \to \mathbf{B}^3 U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^4 U(1) = \mathbf{\flat}_{dR} \mathbf{B}^4 \mathbb{R} \,. \end{displaymath} If we have a $Spin$-principal bundle $X \to \mathbf{B}G$, we may form over it the covering circle $n$-group bundles on which these higher cocycles naturally live \begin{displaymath} \itexarray{ P_2 &\to& \mathbf{B}Fivebrane &\to& * \\ \downarrow && \downarrow && \downarrow \\ P_1 &\to& \mathbf{B}String &\to& \mathbf{B}^7 U(1) &\to& * \\ \downarrow && \downarrow && && \downarrow \\ X &\to& \mathbf{B}Spin &\to& \cdots &\to& \mathbf{B}^3 U(1) } \,. \end{displaymath} Here for $X$ an ordinary space, $X = \tau_0 X$, the higher circle $n$-group principal bundes $P_k$ have the property that also $\tau_0 P_k = X$. Therefore the 0-truncation of the entire composite \begin{displaymath} P_1 \to \mathbf{B}^7 U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^8 \mathbb{R} \end{displaymath} defines a closed 8-form on $X$. This is the curvature characeristic form given by the Chern-Weil homomorphism in this degree. Its refinement to Deligne cohomology in this construction lives naturally not on $X$, but on the covering $P_1$ of $X$. (\ldots{}) \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} \hypertarget{principal_1bundles_with_connection}{}\subsubsection*{{Principal 1-bundles with connection}}\label{principal_1bundles_with_connection} We spell out here how the general theory of \hyperlink{InfinityLieAlgebraConnection}{∞-Lie algebra valued connection} reduces to the standard notion of [[nLab:connection on a bundle|connections]] on ordinary $G$-[[nLab:principal bundle]]s and how the \hyperlink{InfChernWeil}{∞-Chern-Weil homomorphism} reduces on these to the ordinary [[nLab:Chern-Weil homomorphism]]. Let $\mathfrak{g}$ be a (finite dimensional) [[nLab:Lie algebra]]. Then \begin{displaymath} \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G \end{displaymath} is the [[nLab:delooping]] of the simply connected [[nLab:Lie group]] $G$ integrating $\mathfrak{g}$. \begin{uprop} The coefficient object $\mathbf{B}G_{conn}$ of \hyperlink{InfinityLieAlgebraConnection}{genuine ∞-Lie algebra connections} for $\mathfrak{g}$ an ordinary [[nLab:Lie algebra]] is weakly equivalent to the simplicial presheaf \begin{displaymath} \mathbf{B}G_{conn} \stackrel{\simeq}{\to} \Xi[G\times \Omega^1(-,\mathfrak{g}) \stackrel{ \overset{Ad_{p_1}(p_2)+ p_1 d p_1^{-1}}{\to}}{\underset{p_2}{\to}} \Omega^1(-,\mathfrak{g})] \end{displaymath} that assigns objectwise the [[nLab:groupoid of Lie-algebra valued 1-forms]]. This is moreover isomorphic to the simplicial presheaf \begin{displaymath} \cdots = [CartSp^{op},sSet](\mathbf{P}_1(-),\mathbf{B}G]) \end{displaymath} of morphisms out of the [[nLab:path groupoid]]. The flat coefficient object $\mathbf{\flat}\mathbf{B}G$ is modeled by the subobject \begin{displaymath} \Xi[G\times \Omega^1_{flat}(-,\mathfrak{g}) \stackrel{ \overset{Ad_{p_1}(p_2)+ p_1 d p_1^{-1}}{\to}}{\underset{p_2}{\to}} \Omega^1_{flat}(-,\mathfrak{g})] \end{displaymath} of groupoids of Lie-algebra valued forms with vanishing [[nLab:curvature]] 2-form. This is isomorphic to \begin{displaymath} \cdots = [CartSp^{op},sSet](\mathbf{\Pi}_1(-),\mathbf{B}G]) \end{displaymath} of morphism out of the [[nLab:fundamental groupoid]]. \end{uprop} \begin{proof} The statements about morphisms out of the path groupoid are discussed in detail in \href{http://arxiv.org/abs/0705.0452}{SchrWalI}. \end{proof} \begin{ucor} For $X$ a [[nLab:paracompact space|paracompact]] [[nLab:smooth manifold]] and $\{U_i \to X\}$ a [[nLab:good open cover]] we have a natural equivalence of groupoids \begin{displaymath} [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}G_{conn}) \simeq G Bund_\nabla(X) \end{displaymath} with the groupoid of smooth $G$-principal bundles with connection on $G$. For $\langle - \rangle : inn(\mathfrak{g}) \to b^p \mathbb{R}$ an [[nLab:invariant polynomial]] on $\mathfrak{g}$, the \hyperlink{InfChernWeil}{induced morphism} \begin{displaymath} [CartSp^{op}, sSet](C(\U_i\}), \mathbf{B}G_{diff}) \stackrel{\langle F_{(-)}\rangle}{\to} \Omega^{p+1}_{cl}(X) \end{displaymath} is that of the ordinary [[nLab:Chern-Weil homomorphism]]. \end{ucor} We have seen that a refinement of the Chern-Weil homomorphism is available. The above morphism extends to a morphism \begin{displaymath} \langle F_{(-)}\rangle : \mathbf{H}(-, \mathbf{B}G) \to \mathbf{H}(-, \tau_{1} \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R} ) \end{displaymath} in $\mathbf{H} = \infty LieGrpd$ represented by \begin{displaymath} \itexarray{ [CartSp^{op},sSet](-, \mathbf{B}G_{diff}) &\to& [CartSp^{op},sSet](-, \mathbf{cosk}_2 \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R}) \\ \downarrow^{\simeq} \\ [CartSp^{op},sSet](-, \mathbf{B}G) } \,. \end{displaymath} For $X$ a smooth manifold with good cover $\{U_i \to X\}$ we have that $[CartSp^{op},sSet](C\{U_i\}, \mathbf{cosk}_2 \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R})$ is the groupoid whose objects are closed $p+1$-forms on $X$ and whose morphisms are given by $p$-forms modulo exact forms. Let $i \in I$ range over a set of generators for all [[nLab:invariant polynomial]]s. Then \begin{displaymath} \mathbf{H}(-,\mathbf{B}G) \to \prod_i \tau_1\mathbf{\flat}_{dR}\mathbf{B}^{n_i} \mathbb{R} \end{displaymath} is an approximation to the intrinsic Chern-character. We may consider its [[nLab:homotopy fiber]]s over a given set $Q_i$ of curvature characteristic forms. Assume $\nabla, \nabla' : C(\{U_i\}) \to \mathbf{B}G_{diff}$ are two genuine connections with coinciding curvature characteristic classes $\{Q_i\}$. Then in the homotopy fiber they are coboundant cocycles precisely if all the [[nLab:Chern-Simons form]]s $CS_i(\nabla,\nabla')$ vanish modulo an exact form. This equivalence relation is that which defines [[nLab:Simons-Sullivan structured bundle]]s. Their [[nLab:Grothendieck group]] completion yields [[nLab:differential K-theory]]. \hypertarget{principal_2bundles_with_connection}{}\subsubsection*{{Principal 2-bundles with connection}}\label{principal_2bundles_with_connection} (\ldots{}) Let $\mathfrak{g}$ be a Lie [[nLab:strict 2-group]] coming from a [[nLab:differential crossed module]] $(\mathfrak{g}_2 \to \mathfrak{g}_1)$. Then we have two candidate Lie 2-groups integrating this: on the one hand the [[nLab:strict 2-group]] coming from the [[nLab:crossed module]] $(G_2 \to G_1)$ that integrates $(\mathfrak{g}_2 \to\mathfrak{g}_1)$ degreewise as ordinary Lie algebras, and on the other hand $cosk_k\exp(\mathfrak{g})$. \begin{uprop} The morphism \begin{displaymath} \tau_2 \exp(\mathfrak{g}) \to \mathbf{B}(G_2 \to G_1) \end{displaymath} given by evaluating 2-dimensional [[nLab:parallel transport]] is a weak equivalence. \end{uprop} \begin{proof} Use the 3-dimensional nonabelian Stokes theorem from the appendix of \href{http://arxiv.org/abs/0802.0663}{SchrWalII}. \end{proof} \begin{ucor} The object $\mathbf{B}(G_2 \to G_1)_{conn}$ assigns to $U \in CartSp$ the [[nLab:2-groupoid of Lie 2-algebra valued forms]] over $U$. \end{ucor} This is described in detail in \href{http://arxiv.org/abs/0802.0663}{SchrWalII}, subject to the extra constraint that the 2-form curvature vanishes. \begin{ucor} A genuine connection on a $(G_2 \to G_1)$-[[nLab:principal 2-bundle]] with given cocycle $X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}(G_2 \to G_1)$ is a cocycle $X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}(G_2 \to G_1)_{conn}$ given as follows: \begin{enumerate}% \item on $U_i$ a pair of forms $A_i \in \Omega^1(U_i, \mathfrak{g}_1)$, $B_i \in \Omega^2(U_i, \mathfrak{g}_2)$; \end{enumerate} 1 on $U_i \cap U_j$ a function $g_{i j} \in C^\infty(U_{i}\cap U_j , G_1)$ and a 1-form $a_{i j} \in \Omega^1(U_i \cap U_j, \mathfrak{g}_2)$ such that \ldots{} \begin{enumerate}% \item and so forth \end{enumerate} \end{ucor} This is described in detail in \href{}{SchrWalIII}, subject to the extra constraint that the 2-form curvature vanishes. (\ldots{}) \hypertarget{DiffStringStruc}{}\subsubsection*{{Twisted differential $String-$ and $Fivebrane$-structures}}\label{DiffStringStruc} We discuss now in detail refined Chern-Weil morphisms \begin{displaymath} \hat \mathbf{c} : \mathbf{H}_{conn}(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X, \mathbf{B}^n U(1)) \end{displaymath} that send \hyperlink{InfinityLieAlgebraConnection}{∞-connections} on $G$-[[principal ∞-bundle]]s to [[circle n-bundles with connection]] that represent a given [[characteristic class]]. $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ with coefficients in the \hyperlink{http://ncatlab.org/nlab/show/Lie+infinity-groupoid#BnU1}{circle n-groupoid}. Specifically, we consider the first two steps in the of the [[orthogonal group]] $O$ that are controled by [[∞-Lie algebra cohomology]]. The smooth Whitehead tower of $O$ in [[?LieGrpd]] starts as \begin{displaymath} \cdots \to \mathbf{B}Spin \to \mathbf{B} SO \to \mathbf{B}O \,, \end{displaymath} where \begin{itemize}% \item the [[delooping]] $\mathbf{B} SO$ of the [[special orthogonal group]] is the $\mathbb{Z}_2$-[[principal bundle]] over $\mathbf{B}O$ classified by the [[cocycle]] $\mathbf{B}O \to \mathbf{B} \mathbb{Z}_2$ that sends an elemen $k \in O$ to $+1$ if it is in the connected component of the identity and to $-1$ if it is not. This means we have an [[(∞,1)-pullback]] diagram \begin{displaymath} \itexarray{ \mathbf{B} SO &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} O &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B} \mathbb{Z}_2 } \; \end{displaymath} \item the [[delooping]] $\mathbf{B} Spin$ of the [[spin group]] is the is the $\mathbf{B}\mathbb{Z}_2$-[[principal 2-bundle]] over $\mathbf{B} SO$ classified by the [[Stiefel-Whitney class]] $\mathbf{B} SO \to \mathbf{B}^2 \mathbb{Z}$ \begin{displaymath} \itexarray{ \mathbf{B} Spin &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \; \end{displaymath} \end{itemize} Since these two steps are controled by the [[torsion]]-group $\mathbb{Z}_2$ they have no nontrivial refinement to differential cohomology. The next step however is controled by what in the [[(∞,1)-topos]] [[∞Grpd]] $\simeq$ [[Top]] is the first [[fractional Pontryagin class]] $\frac{1}{2}p_1 : \mathcal{B} Spin \to \mathcal{B}^4 \mathbb{Z}$ and which lifts through the [[schreiber:path ∞-groupoid]] functor $\Pi : \infty LieGrpd \to \infty Grpd$ to a characteristic class in $\mathbf{H} =$ [[?LieGrpd]] (as discussed there) $\frac{1}{2} p_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)$ with coefficients in the smooth . This cocycle does arise as the [[Lie integration]] $\exp(\mu)$ of the canonical [[Lie algebra cohomology|Lie algebra 3-cocycle]] $\mu = \langle -,[-,-]\rangle: \mathfrak{so} \to b^2 \mathbb{R}$. The [[principal ∞-bundle|principal 3-bundle]] that this classifies is the [[delooping]] $\mathbf{B} String$ of the [[string 2-group]] $String$ \begin{displaymath} \itexarray{ \mathbf{B} String &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) } \,. \end{displaymath} Notice that the fact that this is an [[(∞,1)-pullback]] implies that for any $X\in \mathbf{H} = \infty LieGrpd$ also \begin{displaymath} \itexarray{ \mathbf{H}(X,\mathbf{B} String) &\to& * \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B} Spin) &\to& \mathbf{H}(X,\mathbf{B}^3 U(1)) } \,, \end{displaymath} which exhibits the [[2-groupoid]] $\mathbf{H}(X,\mathbf{B}String)$ of [[string structure]]s. As we refine in this diagram the bottom morphism to differential cohomology, we obtain correspondingly differential string structures. \hypertarget{StringStructure}{}\paragraph*{{The string-lifting Chern--Simons $3$-bundle with connection}}\label{StringStructure} We describe the special case of the general \hyperlink{InfChernWeil}{$\infty$-Chern--Weil homomorphism} for \hyperlink{InfinityLieAlgebraConnection}{$\infty$-Lie algebra valued connections} corresponding to the [[characteristic class]] $\frac{1}{2}p_1\colon \mathbf{B}Spin \to \mathbf{B}^3 U(1)$: the first fractional [[Pontryagin class]] of the [[spin group]] $\mathbf{B}Spin$. The $\mathbf{B}^3 U(1)$-differential cocycle that it produces from a given $Spin$-[[principal bundle]] is the [[Chern-Simons 2-gerbe|Chern?Simons 2-bundle]] with connection whose class is the obstruction for the existence of a [[string structure]]. The content of this subsection is at [[Chern-Simons 2-gerbe|Chern?Simons 2-gerbe]] in \href{Chern-Simons+2-gerbe#InInfChernWeil}{the section on $\infty$-Chern--Weil theory}. \hypertarget{differential_string_structures}{}\paragraph*{{Differential string structures}}\label{differential_string_structures} The content of this section is at [[differential string structure]]. \hypertarget{FivebraneStructure}{}\paragraph*{{The Fivebrane-lifting Chern-Simons 7-bundle with connection}}\label{FivebraneStructure} The content of this section is at [[Chern-Simons circle 7-bundle]]. \hypertarget{DiffFivebraneStrucs}{}\paragraph*{{Differential fivebrane structures}}\label{DiffFivebraneStrucs} Let \begin{displaymath} \frac{1}{6}\hat p_2 : \mathbf{H}_{conn}(-,\mathbf{B}String) \to \mathbf{H}_{diff}(-, \mathbf{B}^7 U(1)) \end{displaymath} be the differential refinement of the second fractional Pontryagin class discussed \hyperlink{FivebraneStructure}{above}. \textbf{Definition} For $X \in \mathbf{H} =$ [[nLab:?LieGrpd]], the $\infty$-groupoid of \textbf{differential fivebrane-structures} $Fivebrane_{diff}(X)$ is the [[nLab:homotopy fiber]] of $\frac{1}{6}p_2(X) : \mathbf{H}(X,\mathbf{B}String) \to \mathbf{H}_{diff}(X, \mathbf{B}^7 U(1))$. More generally, the $\infty$-groupoid of \textbf{twisted differential fivebrane structures} is the [[nLab:(∞,1)-pullback]] $Fivebrane_{diff,tw}(X)$ in \begin{displaymath} \itexarray{ Fivebrane_{diff,tw}(X) &\to& H_{diff}(X,\mathbf{B}^7 U(1)) \\ \downarrow && \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}String) &\stackrel{\frac{1}{6}\hat p_2}{\to}& \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1)) } \,. \end{displaymath} In terms of the underlying $\infty$-Lie algebra valued local connection data, i.e. before Lie integration in the above sense , this has been considered in (\ldots{}) \hypertarget{chernsimons_theory}{}\subsubsection*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory} The refined higher Chern-Weil homomorphism takes values in [[circle n-bundles with connection]] in [[ordinary differential cohomology]]. Each of these comes with a notion of higher [[holonomy]] over $n$-dimensional curves $\Sigma_n \to X$. The map that takes a [[connection on an infinity-bundle]] to this holonomy is a generalization of the [[action functional]] of [[Chern-Simons theory]]. Therefore the higher Chern-Weil homomorphism defines a class of [[sigma-model]] [[quantum field theories]] that we call [[schreiber:infinity-Chern-Simons theory]]. See there for more details. Special noteworthy cases are \begin{itemize}% \item the class of [[AKSZ sigma-models]] (\hyperlink{FRS}{FRS11}) \item higher [[Chern-Simons supergravity]] (see [[D'Auria-Fre formulation of supergravity]]) \end{itemize} \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[schreiber:differential cohomology in a cohesive topos]] \begin{itemize}% \item \textbf{∞-Chern-Weil theory} \begin{itemize}% \item [[schreiber:Principal ∞-bundles -- models and general theory]] \item [[schreiber:Cech cocycles for differential characteristic classes]] \item [[schreiber:Twisted differential structures]] \begin{itemize}% \item [[schreiber:Twisted Differential String and Fivebrane Structures]] \end{itemize} \end{itemize} \item [[schreiber:∞-Chern-Simons theory]] \begin{itemize}% \item [[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models]] \end{itemize} \end{itemize} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} An explicit presentation of the $\infty$-Chern-Weil homomorphism in terms of [[simplicial presheaves]] and the application to [[differential string structure]]s and [[differential fivebrane structure]]s is considered in \begin{itemize}% \item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech Cocycles for Differential characteristic Classes]]} \end{itemize} The special case that gives the [[AKSZ sigma-model]] is discussed in \begin{itemize}% \item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]} \end{itemize} A general abstract account is in \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} For a commented list of related literature see [[!redirects ∞-Chern-Weil theory]] [[!redirects ∞-Chern-Weil homomorphism]] [[!redirects Chern-Weil homomorphism in Smooth∞Grpd]] [[!redirects infinity-Chern-Weil theory]] \end{document}