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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*{geometry of physics -- WZW terms} \begin{quote}% this entry is one chapter of \emph{[[geometry of physics]]} previous chapters: \emph{[[geometry of physics -- groups|groups]]}, \emph{[[geometry of physics -- principal connections|principal connections]]} next chapter: \emph{[[geometry of physics -- BPS charges|BPS charges]]} \end{quote} \vspace{.5em} \hrule \vspace{.5em} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{wzw_terms}{\textbf{WZW terms}}\dotfill \pageref*{wzw_terms} \linebreak \noindent\hyperlink{model_layer}{Model layer}\dotfill \pageref*{model_layer} \linebreak \noindent\hyperlink{lie_integration}{Lie integration}\dotfill \pageref*{lie_integration} \linebreak \noindent\hyperlink{the_wzw_terms}{The WZW terms}\dotfill \pageref*{the_wzw_terms} \linebreak \noindent\hyperlink{ConsecutiveWZWTermsAndTwists}{Consecutive WZW terms and twists}\dotfill \pageref*{ConsecutiveWZWTermsAndTwists} \linebreak \noindent\hyperlink{semantic_layer}{Semantic layer}\dotfill \pageref*{semantic_layer} \linebreak \noindent\hyperlink{refinement_of_hodge_filtrations}{Refinement of Hodge filtrations}\dotfill \pageref*{refinement_of_hodge_filtrations} \linebreak \noindent\hyperlink{wzw_terms_2}{WZW terms}\dotfill \pageref*{wzw_terms_2} \linebreak \noindent\hyperlink{definite_globalization_of_wzw_terms}{Definite globalization of WZW terms}\dotfill \pageref*{definite_globalization_of_wzw_terms} \linebreak \noindent\hyperlink{syntax_layer}{Syntax layer}\dotfill \pageref*{syntax_layer} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{wzw_terms}{}\subsection*{{\textbf{WZW terms}}}\label{wzw_terms} We had seen that in [[higher differential geometry]], then given any closed [[differential n-form|differential (p+2)-form]] $\omega \in \Omega^{p+2}_{cl}(X)$, it is natural to ask for a [[prequantization]] of it, namely for a [[circle n-bundle with connection|circle (p+1)-bundle with connection]] $\nabla$ (equivalently: [[cocycle]] in degree-$(p+2)$-[[Deligne cohomology]]) on $X$ whose [[curvature]] is $F_\nabla = \omega$. In terms of [[moduli stacks]] this means asking for lifts of the form \begin{displaymath} \itexarray{ && \mathbf{B}^{p+1}U(1)_{conn} \\ &{}^{\mathllap{\nabla}}\nearrow& \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} } \end{displaymath} in the [[homotopy theory]] of [[smooth homotopy types]]. This immediately raises the question for natural classes of examples of such prequantizations. One such class arises in [[infinity-Lie theory]], where $\omega$ is a [[left invariant form]] on a [[smooth infinity-group]] given by a [[cocycle]] in [[L-∞ algebra cohomology]]. The [[prequantum n-bundles]] arising this way are the higher [[WZW terms]] discussed here. In low degree of traditional [[Lie theory]] this appears as follows: On [[Lie groups]] $G$, those closed $(p+2)$-forms $\omega$ which are [[left invariant forms]] may be identified, via the general theory of [[Chevalley-Eilenberg algebras]], with degree $(p+2)$-[[cocycles]] $\mu$ in the [[Lie algebra cohomology]] of the [[Lie algebra]] $\mathfrak{g}$ corresponding to $G$. We have $\omega = \mu(\theta)$where $\theta$ is the [[Maurer-Cartan form]] on $G$. These cocycles $\mu$ in turn may arise, via the [[van Est map]], as the [[Lie differentiation]] of a degree-$(p+2)$-[[cocycle]] $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+2}U(1)$ in the [[Lie group cohomology]] of $G$ itself, with [[coefficients]] in the [[circle group]] $U(1)$. This happens to be the case notably for $G$ a [[simply connected topological space|simply connected]] [[compact Lie group|compact]] [[semisimple Lie group]] such as [[special unitary group|SU]] or [[spin group|Spin]], where $\mu = \langle -,[-,-]\rangle$, hence $\omega = \langle \theta , [\theta,\theta]\rangle$, is the [[Lie algebra cohomology|Lie algebra 3-cocycle]] in [[transgression]] with the [[Killing form]] [[invariant polynomial]] $\langle -,-\rangle$. This is, up to normalization, a representative of the de Rham image of a generator $\mathbf{c}$ of $H^3(\mathbf{B}G, U(1)) \simeq H^4(B G, \mathbb{Z}) \simeq \mathbb{Z}$. Generally, by the discussion at \emph{[[geometry of physics -- principal bundles]]}, the cocycle $\mathbf{c}$ [[modulating morphism|modulates]] an [[infinity-group extension]] which is a [[circle n-group|circle p-group]]-[[principal infinity-bundle]] \begin{displaymath} \itexarray{ \mathbf{B}^p U(1) &\longrightarrow& \hat G \\ && \downarrow \\ && G &\stackrel{\Omega\mathbf{c}}{\longrightarrow}& \mathbf{B}^{p+1}U(1) } \end{displaymath} whose higher [[Dixmier-Douady class]] class $\int \Omega \mathbf{c} \in H^{p+2}(X,\mathbb{Z})$ is an [[integral cohomology]] lift of the [[real cohomology]] class encoded by $\omega$ under the [[de Rham isomorphism]]. In the example of [[spin group|Spin]] and $p = 1$ this extension is the [[string 2-group]]. Such a [[Lie theory|Lie theoretic]] situation is concisely expressed by a diagram of [[smooth homotopy types]] of the form \begin{displaymath} \itexarray{ && &\longrightarrow& \mathbf{B}^{p+1}U(1) \\ &{}^{\mathllap{\Omega \mathbf{c}}}\nearrow& &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}} \\ G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \,, \end{displaymath} where $\flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \simeq \flat_{dR}\mathbf{B}^{p+2}U(1)$ is the \href{cohesive+infinity-topos+--+structures#deRhamCohomology}{de Rham coefficients} (see also at \emph{[[geometry of physics -- de Rham coefficients]]}) and where the homotopy filling the diagram is what exhibits $\omega$ as a de Rham representative of $\Omega \mathbf{c}$. Now, by the very [[homotopy pullback]]-characterization of the [[Deligne complex]] $\mathbf{B}^{p+1}U(1)_{conn}$ (\href{Deligne+cohomology#TheExactDifferentialCohomologyHexagon}{here}), such a diagram is equivalently a [[prequantization]] of $\omega$: \begin{displaymath} \itexarray{ && \mathbf{B}^{p+1}U(1)_{conn} &\longrightarrow& \mathbf{B}^{p+1}U(1) \\ &{}^\mathllap{\nabla}\nearrow& \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}} \\ G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \,. \end{displaymath} For $\omega = \langle -,[-,-]\rangle$ as above, we have $p= 1$ and so $\nabla$ here is a [[circle n-bundle with connection|circle 2-bundle with connection]], often referred to as a [[bundle gerbe]] [[connection on a bundle gerbe|with connection]]. As such, this is also known as the \emph{WZW gerbe} or similar. This terminology arises as follows. In (\href{Wess-Zumino-Witten+model#WessZumino71}{Wess-Zumino 84}) the [[sigma-model]] for a [[string]] propagating on the [[Lie group]] $G$ was considered, with only the standard [[kinetic action]] term. Then in (\href{Wess-Zumino-Witten+model#Witten84}{Witten 84}) it was observed that for this [[action functional]] to give a [[conformal field theory]] after [[quantization]], a certain [[higher gauge theory|higher gauge]] [[interaction term]] has to the added. The resulting [[sigma-model]] came to be known as the \emph{[[Wess-Zumino-Witten model]]} or \emph{WZW model} for short, and the term that Witten added became the \emph{WZW term}. In terms of [[string theory]] it describes the propagation of the [[string]] on the group $G$ subject to a [[force]] of [[gravity]] given by the [[Killing form]] [[Riemannian metric]] and subject to a [[B-field]] [[higher gauge field|higher gauge force]] whose [[field strength]] is $\omega$. In (\href{Wess-Zumino-Witten+model#Gawedzki87}{Gawedzki 87}) it was observed that when formulated properly and generally, this WZW term is the [[surface holonomy]] functional of a [[connection on a bundle gerbe]] $\nabla$ on $G$. This is equivalently the $\nabla$ that we just motivated above. Later, such WZW terms, or at least their curvature forms $\omega$, were recognized all over the place in [[quantum field theory]]. For instance the [[Green-Schwarz sigma-models for super p-branes]] each have an [[action functional]] that is the sum of the standard [[kinetic action]] plus a WZW term of degree $p+2$. In general WZW terms are ``[[gauged WZW model|gauged]]'' which means, as we will see, that they are not defined on the given [[smooth infinity-group]] $G$ itself, but on a bundle $\tilde G$ of [[differential cohomology|differential]] [[moduli stacks]] over that group, such that a map $\Sigma \to \tilde G$ is a pair consisting of a map $\Sigma \to G$ and of a [[higher gauge field]] on $\Sigma$ (a ``tensor multiplet'' of fields). Here we discuss the general construction and theory of such higher WZW terms. \hypertarget{model_layer}{}\subsubsection*{{Model layer}}\label{model_layer} We discuss how every [[cocycle]] $\mu \colon \mathfrak{g} \to b^{p+1} \mathbb{R}$ in [[L-∞ algebra cohomology]] has a [[Lie integration]] to a higher WZW term of the form \begin{displaymath} \mathbf{L}_\mu \colon \tilde G \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} \end{displaymath} where $\tilde G$ is a differential extension of the smooth $p+1$-group that is the universal [[Lie integration]] of $\mathfrak{g}$, and where $\Gamma \hookrightarrow \mathbb{R}$ is the group of [[periods]] of $\mu$ on that group. This construction is a differential refinement of [[Lie integration]], so we start with recalling the relevant constructions and facts of Lie integration. \hypertarget{lie_integration}{}\paragraph*{{Lie integration}}\label{lie_integration} So let $\mathfrak{g}$ be an [[L-∞ algebra]] of [[finite type]]. Write \begin{itemize}% \item $CE(\mathfrak{g})$ for the [[Chevalley-Eilenberg algebra]] of an [[L-∞ algebra]] $\mathfrak{g}$; \item $\Delta^\bullet_{smth} \colon \Delta \to SmoothMfd$ for the [[cosimplicial object|cosimplicial]] [[smooth manifold]] [[manifold with corners|with corners]] which is in degree $k$ the standard $k$-simpliex $\Delta^k \hookrightarrow \mathbb{R}^{k+1}$; \item $\Omega^\bullet_{si}(\Delta_{smth}^k)$ for the [[de Rham complex]] of those [[differential forms]] on $\Delta_{smth}^k$ which have [[sitting instants]], in that in an [[open neighbourhood]] of the [[boundary]] they are constant perpendicular to any face on their value at that face; \item $\Omega^\bullet_{si}(U \times \Delta_{smth}^k)$ for $U \in SmoothMfd$ for the [[de Rham complex]] of differential forms on $U \times \Delta^k$ which when restricted to each point of $U$ have sitting instants on $\Delta^k$; \item $\Omega^\bullet_{vert,si}(U \times \Delta_{smth}^k)$ for the subcomplex of forms that in addition are [[vertical differential forms]] with respect to the projection $U \times \Delta^k \to U$. \end{itemize} \begin{defn} \label{SimplicialLieIntegrationOfLinfinityAlgebra}\hypertarget{SimplicialLieIntegrationOfLinfinityAlgebra}{} For $\mathfrak{g}$ an [[L-∞ algebra]], write \begin{itemize}% \item $\exp(\mathfrak{g})_\bullet \in PreSmoothTypes = PSh(CartSp,sSet)$ for the [[simplicial presheaf]] \begin{displaymath} \exp(\mathfrak{g}) \colon (U \times k) \mapsto Hom_{dgAlg}( CE(\mathfrak{g}), \Omega^\bullet_{vert,si}(U \times \Delta_{smth}^k) ) \,. \end{displaymath} which is the universal \emph{[[Lie integration]]} of $\mathfrak{g}$; \item $\flat_{dR}\exp(\mathfrak{g})_\bullet \in PreSmoothTypes = PSh(CartSp,sSet)$ for the [[simplicial presheaf]] \begin{displaymath} \flat_{dR}\exp(\mathfrak{g})_\bullet \;\colon\; (U \times k) \mapsto Hom_{dgAlg}( CE(\mathfrak{g}), \Omega^{\bullet\geq 1, \bullet}_{si}(U \times \Delta^k_{smth}) ) \end{displaymath} of those differential forms on $U \times \Delta^\bullet$ with at least one leg along $U$; \item $\Omega^1_{flat}(-,\mathfrak{g}) \coloneqq \flat_{dR}\exp(\mathfrak{g})_0 \longrightarrow \flat_{dR}\exp(\mathfrak{g})_\bullet$ for the canonical inclusion of the degree-0 piece, regarded as a simplicial constant simplicial presheaf. \end{itemize} \end{defn} \begin{example} \label{ExamplesOfLieIntegration}\hypertarget{ExamplesOfLieIntegration}{} From the discussion at \emph{[[Lie integration]]}: \begin{enumerate}% \item $\Omega^1_{flat}(-,b^{p+1}\mathbb{R}) = \mathbf{\Omega}^{p+2}_{cl}$; \item for $\mathfrak{g}$ an ordinary [[Lie algebra]], then for the [[simplicial skeleton|2-coskeleton]] \begin{displaymath} cosk_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G_\bullet \end{displaymath} for $G$ the simply connected [[Lie group]] associated to $\mathfrak{g}$ by traditional [[Lie theory]]. If $\mathfrak{g}$ is furthermore a [[semisimple Lie algebra]], then also \begin{displaymath} cosk_3 \exp(\mathfrak{g}) \simeq \mathbf{B}G_\bullet \end{displaymath} \item for $\mathfrak{g} = b^{p}\mathbb{R}$ the [[line Lie n-algebra|line Lie p+1]]-algebra, then \begin{displaymath} \exp(b^p \mathbb{R}) \simeq \mathbf{B}^{p+1}\mathbb{R} \,. \end{displaymath} \end{enumerate} \end{example} \begin{remark} \label{LieIntegrationIsFunctorial}\hypertarget{LieIntegrationIsFunctorial}{} The constructions in def. \ref{SimplicialLieIntegrationOfLinfinityAlgebra} are clearly [[functor|functorial]]: given a [[homomorphism]] of [[L-∞ algebras]] \begin{displaymath} \mu \;\colon\; \mathfrak{g} \longrightarrow \mathfrak{h} \end{displaymath} it prolongs to a homomorphism of presheaves \begin{displaymath} \mu \colon \Omega^1_{flat}(-,\mathfrak{g}) \longrightarrow \Omega^1(-,\mathfrak{h}) \end{displaymath} and of [[simplicial presheaves]] \begin{displaymath} \exp(\mu) \;\colon\; \exp(\mathfrak{g}) \longrightarrow \exp(\mathfrak{h}) \end{displaymath} etc. \end{remark} \begin{example} \label{LInfinityCocyclesAsMorphisms}\hypertarget{LInfinityCocyclesAsMorphisms}{} A degree-$(p+2)$-[[L-∞ cocycle]] $\mu$ on an [[L-∞ algebra]] $\mathfrak{g}$ is a homomorphism of the form \begin{displaymath} \mu \colon \mathfrak{g} \longrightarrow b^{p+1}\mathbb{R} \end{displaymath} to the [[line Lie n-algebra|line Lie (p+2)-algebra]] $b^{p+1}\mathbb{R}$. The [[formal dual]] of this is the homomorphism of [[dg-algebras]] \begin{displaymath} CE(\mathfrak{g}) \longleftarrow CE(b^{p+1}\mathbb{R}) \colon \mu^\ast \end{displaymath} which manifestly picks a $d_{CE(\mathfrak{g})}$-closed element in $CE^{p+2}(\mathfrak{g})$. Precomposing this $\mu^\ast$ with a flat [[L-∞ algebra valued differential form]] \begin{displaymath} A \in \Omega^1_{flat}(X,\mathfrak{g}) = Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(X)) \end{displaymath} yields, by example \ref{ExamplesOfLieIntegration}, a plain closed $(p+2)$-form \begin{displaymath} \mu^\ast A \in \Omega^{p+2}_{cl}(X) \,. \end{displaymath} \end{example} \begin{defn} \label{GroupOfPeriods}\hypertarget{GroupOfPeriods}{} Given an [[L-∞ cocycle]] \begin{displaymath} \mu \colon \mathfrak{g} \longrightarrow b^{p+1}\mathbb{R} \,, \end{displaymath} as in example \ref{LInfinityCocyclesAsMorphisms}, then its \emph{group of [[periods]]} is the [[discrete group|discrete]] additive [[subgroup]] $\Gamma \hookrightarrow \mathbb{R}$ of those [[real numbers]] which are [[integration of differential forms|integrations]] \begin{displaymath} \int_{\partial \Delta^{p+3}_{smth}} \mu^\ast A \in \mathbb{R} \end{displaymath} of the value of $\mu$, as in example \ref{LInfinityCocyclesAsMorphisms}, on [[L-∞ algebra valued differential forms]] \begin{displaymath} A \in \Omega^1_{flat}(\partial \Delta^{p+3}_{smth}) \,, \end{displaymath} over the [[boundary of a simplex|boundary of the (p+3)-simplex]] (which are forms with sitting instants on the $(p+2)$-dimensional faces that glue together; without restriction of generality we may simply consider forms on the $(p+2)$-[[sphere]] $S^{p+2}$). \end{defn} \begin{prop} \label{TruncatedLieIntegrationOfCocycle}\hypertarget{TruncatedLieIntegrationOfCocycle}{} Given an [[L-∞ cocycle]] $\mu \colon \mathfrak{g} \to b^{p+1}\mathbb{R}$, as in example \ref{LInfinityCocyclesAsMorphisms}, then the universal Lie integration of $\mu$, via def. \ref{SimplicialLieIntegrationOfLinfinityAlgebra} and remark \ref{LieIntegrationIsFunctorial}, descends to the $(p+2)$-[[coskeleton]] \begin{displaymath} \mathbf{B}G_\bullet \coloneqq cosk_{p+2}\exp(\mathfrak{g}) \end{displaymath} up to quotienting the coefficients $\mathbb{R}$ by the group of [[periods]] $\Gamma$ of $\mu$, def. \ref{GroupOfPeriods}, to yield the bottom morphism in \begin{displaymath} \itexarray{ \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\longrightarrow}& \mathbf{B}^{p+2}\mathbb{R} \\ \downarrow && \downarrow \\ \mathbf{B}G_\bullet &\stackrel{\mathbf{c}}{\longrightarrow}& \mathbf{B}^{p+2} (\mathbb{R}/\Gamma)_\bullet } \,. \end{displaymath} \end{prop} This is fairly immediate from the definitions, detailed discussion is in (\href{Lie+integration#FSS12}{FSS 12}). Here and in the following we are freely using example \ref{ExamplesOfLieIntegration} to identify $\exp(b^{p+1}\mathbb{R}) \simeq \mathbf{B}^{p+2}\mathbb{R}$. Establishing this is the only real work in prop. \ref{TruncatedLieIntegrationOfCocycle}. \hypertarget{the_wzw_terms}{}\paragraph*{{The WZW terms}}\label{the_wzw_terms} \begin{prop} \label{HodgeFiltrationRefinementFromLInfinityCocycles}\hypertarget{HodgeFiltrationRefinementFromLInfinityCocycles}{} For $\mu \colon \mathfrak{g}\longrightarrow b^{p+1}\mathbb{R}$ an [[L-∞ cocycle]], then there is the following canonical [[commuting diagram]] of [[simplicial presheaves]] \begin{displaymath} \itexarray{ \Omega^1_{flat}(-,G) &\stackrel{\mu}{\longrightarrow}& \Omega^2_{cl}(-,\mathbb{G}) \\ \downarrow && \downarrow \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR} \mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^2 \mathbb{G} } \;\;\; \coloneqq \;\;\; \itexarray{ \Omega^1_{flat}(-,\mathfrak{g}) &\stackrel{\mu}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} \\ \downarrow && \downarrow \\ \flat_{dR}\exp(\mathfrak{g})_\bullet & \stackrel{\flat_{dR}\exp(\mu)}{\longrightarrow} & \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \\ \downarrow && \downarrow \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \end{displaymath} which is given \begin{itemize}% \item on the top by def. \ref{SimplicialLieIntegrationOfLinfinityAlgebra}, example \ref{ExamplesOfLieIntegration}, remark \ref{LieIntegrationIsFunctorial}, \item on the bottom by prop. \ref{TruncatedLieIntegrationOfCocycle}, \end{itemize} Moreover, this presents a refinement of the canonical [[Hodge filtration]] on $\mathbf{B}^{p}(\mathbb{R}/\Gamma)$, def. \ref{RefinementOfHodgeFiltration}, along the cocycle $\mathbf{c}$ which Lie integrates $\mu$ via prop. \ref{TruncatedLieIntegrationOfCocycle}. \end{prop} \begin{defn} \label{DifferentiallyTiwstedGroup}\hypertarget{DifferentiallyTiwstedGroup}{} Write \begin{displaymath} \tilde G \coloneqq G \underset{\flat_{dR}\mathbf{B}G}{\times} \mathbf{\Omega}^1_{flat}(-,\mathfrak{g}) \end{displaymath} for the [[homotopy pullback]] of the left vertical morphism in prop. \ref{HodgeFiltrationRefinementFromLInfinityCocycles} along (the [[modulating morphism]] for) the [[Maurer-Cartan form]] $\theta_G$ of $G$, i.e. for the object sitting in a homotopy Cartesian square of the form \begin{displaymath} \itexarray{ \tilde G &\stackrel{\theta_{\tilde G}}{\longrightarrow}& \mathbf{\Omega}^1_{flat}(-,\mathfrak{g}) \\ \downarrow && \downarrow \\ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR}\mathbf{B}G } \,. \end{displaymath} \end{defn} \begin{example} \label{ExamplesForTildeGInModLayer}\hypertarget{ExamplesForTildeGInModLayer}{} For the special case that $G$ is an ordinary [[Lie group]], then $\flat_{dR}\mathbf{B}G \simeq \Omega^{1}_{flat}(-,\mathfrak{g})$, hence in this case the morphism being pulled back in def. \ref{DifferentiallyTiwstedGroup} is an [[equivalence]], and so in this case nothing new happens, we get $\tilde G \simeq G$. On the other extreme, when $G = \mathbf{B}^{p}U(1)$ is the [[circle n-group|circle (p+1)-group]], then def. \ref{DifferentiallyTiwstedGroup} reduces to the [[homotopy pullback]] that characterizes the [[Deligne complex]] and hence yields \begin{displaymath} \widetilde{\mathbf{B}^p U(1)} \simeq \mathbf{B}^p U(1)_{conn} \,. \end{displaymath} This shows that def. \ref{DifferentiallyTiwstedGroup} is a certain non-abelian generalization of [[ordinary differential cohomology]]. We find further characterization of this below in corollary \ref{TildeHatGIsDifferentialModuliBundle}, see remark \ref{InterpretationOfTildeHatG}. \end{example} \begin{remark} \label{}\hypertarget{}{} From example \ref{ExamplesForTildeGInModLayer} one see the conceptial meaning of def. \ref{DifferentiallyTiwstedGroup}: For $G$ a [[Lie group]], then the de Rham coefficients are just globally defined differential forms, $\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})$ (by the discussion \href{smooth+infinity-groupoid+--+structures#deRhamWithCoefficientsInBOfLieGroup}{here}), and in particular therefore the [[Maurer-Cartan form]] $\theta_G \colon G \to \flat_{dR}\mathbf{B}G$ is a globally defined differential form. This is no longer the case for general [[smooth ∞-groups]] $G$. In general, the [[Maurer-Cartan forms]] here is a [[cocycle]] in [[hypercohomology]], given only locally by differential forms, that are glued nontrivially, in general, via [[gauge transformations]] and [[higher gauge transformations]] given by lower degree forms. But the WZW terms that we are after are supposed to be [[prequantizations]] of globally defined Maurer-Cartan forms. The homotopy pullback in def. \ref{DifferentiallyTiwstedGroup} is precisely the [[universal construction]] that enforces the existence of a globally defined Maurer-Cartan form for $G$, namely $\theta_{\tilde G} \colon \tilde G \to \Omega^1_{flat}(-,\mathfrak{g})$. \end{remark} \begin{defn} \label{WZWTermFromLieIntegration}\hypertarget{WZWTermFromLieIntegration}{} Given an [[nLab:L-∞ cocycle]] $\mu \colon \mathfrak{g}\longrightarrow b^{p+1}\mathbb{R}$, then via prop. \ref{HodgeFiltrationRefinementFromLInfinityCocycles}, prop. \ref{TruncatedLieIntegrationOfCocycle} and using the [[nLab:natural transformation|naturality]] of the [[nLab:Maurer-Cartan form]], we have a morphism of [[nLab:cospan]] [[nLab:diagrams]] of the form \begin{displaymath} \itexarray{ \Omega^1_{flat}(-,\mathfrak{g}) &\stackrel{\mu}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} \\ \downarrow && \downarrow \\ \flat_{dR} \mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \\ \uparrow^{\mathrlap{\theta_G}} && \uparrow^{\mathrlap{\theta_{\mathbf{B}^{p+1}(\mathbb{R}/\Gamma)}}} \\ G &\stackrel{\Omega \mathbf{c}}{\longrightarrow}& \mathbf{B}^{p+1} (\mathbb{R}/\Gamma) } \,. \end{displaymath} By the homotopy fiber product characterization of the [[nLab:Deligne complex]] this yields a morphism of the form \begin{displaymath} \mathbf{L}_{WZW}^{\mu} \;\colon\; \tilde G \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} \,. \end{displaymath} which [[nLab:modulating morphisms|modulates]] a [[nLab:circle n-bundle with connection|p+1-connection]]/[[nLab:Deligne cohomology|Deligne cocycle]] on the differentially extended smooth $\infty$-group $\tilde G$ from def. \ref{DifferentiallyTiwstedGroup}. This we call the \emph{WZW term} obtained by universal Lie integration from $\mu$. \end{defn} Essentially this construction originates in ([[schreiber:The brane bouquet|FSS 13]]). \begin{remark} \label{}\hypertarget{}{} The WZW term of def. \ref{WZWTermFromLieIntegration} is a [[prequantization]] of \begin{displaymath} \omega \coloneqq \mu(\theta_{\tilde G}) \end{displaymath} \begin{displaymath} \itexarray{ && \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} \\ & {}^{\mathllap{\mathbf{L}_{WZW}^\mu}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ \tilde G &\stackrel{\mu(\theta_{\tilde G})}{\longrightarrow}& \mathbf{\Omega}^{p+2} } \,. \end{displaymath} \end{remark} \hypertarget{ConsecutiveWZWTermsAndTwists}{}\paragraph*{{Consecutive WZW terms and twists}}\label{ConsecutiveWZWTermsAndTwists} More generally, one has a sequence of [[L-∞ cocycles]], each defined on the extension that is classified by the previous one -- a [[schreiber:The brane bouquet|bouquet]] of cocycles. In each stage, for $\mu_1 \colon \mathfrak{g}\to b^{p_1+1}\mathbb{R}$ a cocycle and $\hat {\mathfrak{g}} \to \mathfrak{g}$ the extension that it classifies (its [[homotopy fiber]]), then the next cocycle is of the form $\mu_2 \colon \hat \mathfrak{g} \to b^{p_2+1}\mathbb{R}$ \begin{displaymath} \itexarray{ \hat {\mathfrak{g}} &\stackrel{\mu_2}{\longrightarrow}& b^{p_2+1}\mathbb{R} \\ \downarrow \\ \mathfrak{g} &\stackrel{\mu_1}{\longrightarrow}& b^{p_1+1}\mathbb{R} } \,. \end{displaymath} \begin{lemma} \label{CharacterizationOfLInfinityHomotopyFibers}\hypertarget{CharacterizationOfLInfinityHomotopyFibers}{} The [[homotopy fiber]] $\hat \mathfrak{g} \to \mathfrak{g}$ of $\mu_1$ is given by the ordinary [[pullback]] \begin{displaymath} \itexarray{ \hat \mathfrak{g} &\longrightarrow& e b^{p_1} \mathbb{R} \\ \downarrow && \downarrow \\ \mathfrak{g} &\stackrel{\mu_1}{\longrightarrow}& b^{p_1+1}\mathbb{R} } \,, \end{displaymath} where $e b^{p_1}\mathbb{R}$ is defined by its [[Chevalley-Eilenberg algebra]] $CE(e b^{p_1}\mathbb{R})$ being the [[Weil algebra]] of $b^{p_1}\mathbb{R}$, which is the [[free construction|free]] differential graded algebra on a generator in degree $p_1$, and where the right vertical map takes that generator to 0 and takes its free image under the differential to the generator of $CE(b^{p_1+1}\mathbb{R})$. \end{lemma} \begin{proof} This follows with the \href{model+structure+for+L-infinity+algebras#HomotopyFiberProducts}{recognition principle for L-∞ homotopy fibers}. \end{proof} \begin{cor} \label{PullbackOfDifferentialFormCoefficients}\hypertarget{PullbackOfDifferentialFormCoefficients}{} A [[homotopy fiber sequence]] of [[L-∞ algebras]] $\hat \mathfrak{g} \to \mathfrak{g}\stackrel{\mu}{\longrightarrow} b^{p+1}\mathbb{R}$ induces a [[homotopy pullback]] diagram of the the associated objects of [[L-∞ algebra valued differential forms]], def. \ref{SimplicialLieIntegrationOfLinfinityAlgebra}, of the form \begin{displaymath} \itexarray{ \mathbf{\Omega}^1_{flat}(-,\hat {\mathfrak{g}}) &\stackrel{}{\longrightarrow}& \mathbf{\Omega}^{p+1} \\ \downarrow && \downarrow^{\mathbf{d}} \\ \mathbf{\Omega}^1_{flat}(-,\mathfrak{g}) &\stackrel{\mu}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} } \end{displaymath} (hence an ordinary [[pullback]] of [[presheaves]], since these are all simplicially constant). \end{cor} \begin{proof} The construction $\mathfrak{g} \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(-))$ preserves [[pullbacks]] ($CE$ is an anti-equivalence onto its image, pullbacks of (pre-)-sheaves are computed objectwise, the [[hom-functor]] preserves pullbacks in the covariant argument). Observe then (see the discussion at \emph{[[L-∞ algebra valued differential forms]]), that while} \begin{displaymath} \mathbf{\Omega}^{p+2}_{cl} \simeq Hom_{dgAlg}(CE(b^{p+1}), \Omega^\bullet(-)) \end{displaymath} we have \begin{displaymath} \mathbf{\Omega}^{p+1} \simeq Hom_{dgAlg}(W(b^{p}), \Omega^\bullet(-)) \,. \end{displaymath} With this the statement follows by lemma \ref{CharacterizationOfLInfinityHomotopyFibers}. \end{proof} \begin{defn} \label{ConsecutiveCocycles}\hypertarget{ConsecutiveCocycles}{} We say that a pair of [[L-∞ cocycles]] $(\mu_1, \mu_2)$ is \emph{consecutive} if the domain of the second is the extension ([[homotopy fiber]]) defined by the first \begin{displaymath} \itexarray{ \hat {\mathfrak{g}} &\stackrel{\mu_2}{\longrightarrow}& b^{p_2+1} \\ \downarrow \\ \mathfrak{g} &\stackrel{\mu_1}{\longrightarrow}& b^{p_1+1}\mathbb{R} } \end{displaymath} and if the truncated [[Lie integrations]] of these cocycles via prop. \ref{TruncatedLieIntegrationOfCocycle} preserves the extension property in that also \begin{displaymath} \hat G \to G \stackrel{\Omega \mathbf{c}_1}{\longrightarrow} \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1) \end{displaymath} is a [[homotopy fiber sequence]] of [[smooth homotopy types]]. \end{defn} \begin{remark} \label{}\hypertarget{}{} The issue of the second clause in def. \ref{ConsecutiveCocycles} is to do with the truncation degrees: the universal untruncated [[Lie integration]] $\exp(-)$ preserves homotopy fiber sequences, but if there are non-trivial cocycles on $\mathfrak{g}$ in between $\mu_1$ and $\mu_2$, for $p_2 \gt p_1$, then these will remain as nontrivial homotopy groups in the higher-degree truncation $\mathbf{B}G_{2} \coloneqq \tau_{p_2}\exp(\hat\mathfrak{g})$ (see \href{Lie+integration#Henriques}{Henriques 06, theorem 6.4}) but they will be truncated away in $\mathbf{B}G_1 \coloneqq \tau_{p_1}\exp(\mathfrak{g})$ and will hence spoil the preservation of the homotopy fibers through Lie integration. Notice that extending along consecutive cocycles is like the extension stages in a [[Whitehead tower]]. \end{remark} Given two consecutive [[L-∞ cocycles]] $(\mu_1,\mu_2)$, def. \ref{ConsecutiveCocycles}, let \begin{displaymath} \mathbf{L}_1 \colon \tilde G \longrightarrow \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn} \end{displaymath} and \begin{displaymath} \mathbf{L}_2 \colon \widetilde {\hat G} \longrightarrow \mathbf{B}^{p_2+1}(\mathbb{R}/\Gamma_2)_{conn} \end{displaymath} be the WZW terms obtained from the two cocycles via def. \ref{WZWTermFromLieIntegration}. \begin{prop} \label{ConsecutiveWZWTermsFromConsecutiveLInfinityCocycles}\hypertarget{ConsecutiveWZWTermsFromConsecutiveLInfinityCocycles}{} There is a [[homotopy pullback]] square in [[smooth homotopy types]] of the form \begin{displaymath} \itexarray{ \widetilde {\hat G} &\stackrel{}{\longrightarrow}& \mathbf{\Omega}^{p_1+1} \\ \downarrow && \downarrow \\ \tilde G &\stackrel{\mathbf{L}_1}{\longrightarrow}& \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn} } \,. \end{displaymath} \end{prop} \begin{proof} Consider the following [[pasting]] composite \begin{displaymath} \itexarray{ \mathbf{\Omega}^{p_1+1} &\longrightarrow& \ast &\longleftarrow& \ast \\ {}^{\mathllap{\mathbf{d}}}\downarrow &\swArrow& \downarrow && \downarrow \\ \mathbf{\Omega}^{p_1+2} &\longrightarrow& \flat_{dR}\mathbf{B}^{p_1+2}\mathbb{R} &\stackrel{\theta_{\mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)}}{\longleftarrow}& \mathbf{B}^{p_1+1}\mathbb{R} \\ \uparrow^{\mathrlap{\mu_1}} && \uparrow^{\mathrlap{\flat_{dR} \mathbf{B}G}} && \uparrow^{\mathrlap{\Omega \mathbf{c}_1}} \\ \mathbf{\Omega}^1_{flat}(-,\mathfrak{g}) &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}G &\stackrel{\theta_G}{\longleftarrow}& G } \,, \end{displaymath} where \begin{itemize}% \item the top left square is the evident homotopy; \item the bottom left square is from prop. \ref{HodgeFiltrationRefinementFromLInfinityCocycles} \item the right square is the naturality of the [[Maurer-Cartan form]] construction. \end{itemize} Under forming [[homotopy limits]] over the \emph{horizontal} cospan diagrams here, this turns into \begin{displaymath} \itexarray{ \mathbf{\Omega}^{p_1+1} \\ \downarrow \\ \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn} \\ \uparrow^{\mathrlap{\mathbf{L}_1}} \\ \tilde G } \end{displaymath} by prop. \ref{WZWTermByUniversality}. On the other hand, forming homotopy limits \emph{vertically} this turns into \begin{displaymath} \itexarray{ \mathbf{\Omega}^1_{flat}(-,\hat \mathfrak{g}) &\longrightarrow& \flat_{dR}\mathbf{B}G_2 &\stackrel{\theta_{\hat G}}{\longleftarrow}& \hat G } \end{displaymath} (on the left by corollary \ref{PullbackOfDifferentialFormCoefficients}, on the right by the second clause in def. \ref{ConsecutiveCocycles}). The homotopy limit over that last [[cospan]], in turn, is $\widetilde{\hat G}$. This implies the claim by the fact that homotopy limits commute with each other. \end{proof} \begin{remark} \label{}\hypertarget{}{} Prop. \ref{ConsecutiveWZWTermsFromConsecutiveLInfinityCocycles} says how consecutive pairs of $L_\infty$-cocycles Lie integrate suitably to consecutive pairs of WZW terms. \end{remark} \begin{cor} \label{TildeHatGIsDifferentialModuliBundle}\hypertarget{TildeHatGIsDifferentialModuliBundle}{} In the above situation there is a [[homotopy fiber sequence]] of the form \begin{displaymath} \itexarray{ \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn} &\longrightarrow& \widetilde {\hat G} \\ && \downarrow \\ && \tilde G } \,. \end{displaymath} \end{cor} \begin{proof} By prop. \ref{ConsecutiveWZWTermsFromConsecutiveLInfinityCocycles} and the [[pasting law]], the [[homotopy fiber]] of $\widetilde {\hat G} \to \tilde G$ is equivalently the homotopy fiber of $\mathbf{\Omega}^{p_1+1}\to \mathbf{\Omega}^{p_1+2}_{cl}$ \begin{displaymath} \itexarray{ \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn} &\longrightarrow& \widetilde {\hat G} &\stackrel{}{\longrightarrow}& \mathbf{\Omega}^{p_1+1} \\ \downarrow && \downarrow && \downarrow \\ \ast &\longrightarrow& \tilde G &\stackrel{\mathbf{L}_1}{\longrightarrow}& \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn} } \,. \end{displaymath} \end{proof} \begin{remark} \label{InterpretationOfTildeHatG}\hypertarget{InterpretationOfTildeHatG}{} Corollary \ref{TildeHatGIsDifferentialModuliBundle} says that $\widetilde {\hat G}$ is a [[bundle]] of [[moduli stacks]] for [[differential cohomology]] over $\tilde G$. This means that maps \begin{displaymath} \Sigma \longrightarrow \widetilde{\hat G} \end{displaymath} (which are the [[field (physics)|fields]] of the higher [[WZW model]] with WZW term $\mathbf{L}_2$) are pairs of plain maps $\phi \colon \Sigma \to \tilde G$ together with a differential cocycle on $\Sigma$, i.e. a $p_1$-form connection on $\Sigma$, which is twisted by $\phi$ in a certain way. This oocurs for the (properly globalized) [[Green-Schwarz super p-brane sigma models]] of all the [[D-branes]] and of the [[M5-brane]]. For the D-branes $p_1 = 1$ and so there is a 1-form connection on their [[worldvolume]], the [[Chan-Paton gauge field]]. For the [[M5-brane]] $p_1 = 2$ and so there is a 2-form connection on its worldvolume, the [[self-dual higher gauge field]] in 6d. \end{remark} \hypertarget{semantic_layer}{}\subsubsection*{{Semantic layer}}\label{semantic_layer} We discuss the general abstract formulation of WZW terms in a [[cohesive (infinity,1)-topos]]. Throughout, let \begin{itemize}% \item $\mathbf{H}$ an [[cohesive (∞,1)-topos]]; \item $\mathbb{G} \in \mathbf{H}$ be an object equipped with the structure of a [[braided ∞-group]], i.e. with specified double [[delooping]] $\mathbf{B}^2 \mathbb{G}$. \item $\Omega^2_{cl}(-,\mathbb{G}) \to \cdots \to \flat_{dR}\mathbf{B}\mathbb{G}$ a chosen [[Hodge filtration]]; \item $G \in \mathbf{H}$ be any object equipped with [[∞-group]] structure, i.e. with specified [[delooping]] $\mathbf{B}G$; \item $\mathbf{c} \;\colon\; \mathbf{B}G \longrightarrow \mathbf{B}^2 \mathbb{G}$ a morphism, hence a [[cocycle]] in the [[group cohomology]] of $G$ with [[coefficients]] in $\mathbb{G}$. \end{itemize} Write \begin{itemize}% \item $\mathbf{B}\mathbb{G}_{conn} \coloneqq \mathbf{B}\mathbb{G} \underset{\flat_{dR}\mathbf{B}^2 \mathbb{G}}{\times} \Omega^2_{cl}(-,\mathbb{G})$ for the induced [[coefficients]] for $\mathbb{G}$-[[differential cohomology]], as discussed at \emph{[[geometry of physics -- principal connections]]}; \item $\hat G \to G$ for the [[infinity-group extension]] classified by $\mathbf{c}$. \end{itemize} \hypertarget{refinement_of_hodge_filtrations}{}\paragraph*{{Refinement of Hodge filtrations}}\label{refinement_of_hodge_filtrations} \begin{defn} \label{RefinementOfHodgeFiltration}\hypertarget{RefinementOfHodgeFiltration}{} A \emph{refinement of the [[Hodge filtration]]} of $\mathbb{G}$ along the cocycle $\mathbf{c}$ is a choice of [[0-truncated]] object $\Omega^1_{flat}(-,G) \in \mathbf{H}$ and a completion to a diagram \begin{displaymath} \itexarray{ \Omega^1_{flat}(-,G) &\stackrel{\mu}{\longrightarrow}& \Omega^2_{cl}(-,\mathbb{G}) \\ \downarrow && \downarrow \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR} \mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^2 \mathbb{G} } \end{displaymath} We write $\tilde G$ for the [[homotopy pullback]] of this refinement along the [[Maurer-Cartan form]] $\theta_G$ of $G$ \begin{displaymath} \itexarray{ \tilde G &\stackrel{\theta_{\tilde G}}{\longrightarrow}& \mathbf{\Omega}^1_{flat}(-,G) \\ \downarrow && \downarrow \\ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR}\mathbf{B}G } \,. \end{displaymath} \end{defn} \begin{example} \label{}\hypertarget{}{} Let $\mathbf{H} =$ [[Smooth∞Grpd]] and $\mathbb{G} = \mathbf{B}^p U(1)$ the [[circle n-group|circle (p+1)-group]]. For $G$ an ordinary [[Lie group]], then $\mu$ may be taken to be the [[Lie algebra cohomology|Lie algebra cocycle]] corresponding to $\mathbf{c}$ and then $\tilde G \simeq G$. On the opposite extreme, for $G = \mathbf{B}^p U(1)$ itself with $\mathbf{c}$ the identity, then $\tilde G = \mathbf{B}^pU (1)_{conn}$ is the [[coefficients]] for [[ordinary differential cohomology]] (the [[Deligne complex]] under [[Dold-Kan correspondence]] and [[infinity-stackification]]). Hence a more general case is a fibered product of these two, where $\tilde G$ is such that a map $\Sigma \longrightarrow \tilde G$ is equivalently a pair consisting of a map $\Sigma \to G$ and of differential $p$-form data on $\Sigma$. This is the case of relevance for WZW models of [[super p-branes]] with ``tensor multiplet'' fields on them, such as the [[D-branes]] and the [[M5-brane]]. \end{example} \hypertarget{wzw_terms_2}{}\paragraph*{{WZW terms}}\label{wzw_terms_2} \begin{prop} \label{WZWTermByUniversality}\hypertarget{WZWTermByUniversality}{} In the situation of def. \ref{RefinementOfHodgeFiltration} there is an essentially unique [[prequantum n-bundle|prequantization]] \begin{displaymath} \mathbf{L}_{WZW} \colon \tilde G \longrightarrow \mathbf{B}^2 \mathbb{G}_{conn} \end{displaymath} of the closed differential form \begin{displaymath} \mu(\theta_{\tilde G}) \colon \tilde G \stackrel{\theta_{\tilde G}}{\longrightarrow} \mathbf{\Omega}^1_{flat}(-,G) \stackrel{\mu}{\longrightarrow} \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) \end{displaymath} whose underlying $\mathbb{G}$-[[principal ∞-bundle]] is [[modulating morphism|modulated]] by the [[looping and delooping|looping]] $\Omega \mathbf{c}$ of the original cocycle. This we call the \emph{WZW term} of $\mathbf{c}$ with respect to the chosen refinement of the Hodge structure. \end{prop} \begin{proof} The morphism in question is the image under forming [[homotopy limits]] of the morphism of [[cospan]] diagrams \begin{displaymath} \itexarray{ \Omega^1_{flat}(-,G) &\stackrel{\mu}{\longrightarrow}& \Omega^2_{cl}(-,\mathbb{G}) \\ \downarrow && \downarrow \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^2 \mathbb{G} \\ \uparrow^{\mathrlap{\theta_G}} && \uparrow^{\mathrlap{\theta_{\mathbf{B}\mathbb{G}}}} \\ G &\stackrel{\Omega \mathbf{c}}{\longrightarrow}& \mathbf{B}\mathbb{G} } \,, \end{displaymath} where the top square is from def. \ref{RefinementOfHodgeFiltration} and where the bottom square is the naturality square of the [[homotopy fiber sequence]] that defines the [[Maurer-Cartan forms]] (see \href{Maurer-Cartan+form#OnCohesiveHomotopyTypes}{here}). \end{proof} \hypertarget{definite_globalization_of_wzw_terms}{}\paragraph*{{Definite globalization of WZW terms}}\label{definite_globalization_of_wzw_terms} \ldots{}[[definite globalization of WZW terms]]\ldots{} \hypertarget{syntax_layer}{}\subsubsection*{{Syntax layer}}\label{syntax_layer} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{https://dl.dropboxusercontent.com/u/12630719/dcct.pdf}{pdf}) \end{itemize} \end{document}