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\begin{document}

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\section*{geometry of physics -- BPS charges}

\begin{quote}%
this page is one chapter of \emph{[[geometry of physics]]}

previous chapters: \emph{[[geometry of physics -- manifolds and orbifolds|manifolds and orbifolds]]}, \emph{[[geometry of physics -- WZW terms|WZW terms]]}, \emph{[[geometry of physics -- supergeometry|supergeometry]]}


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{motivation_and_results}{Motivation and results}\dotfill \pageref*{motivation_and_results} \linebreak
\noindent\hyperlink{prerequisites}{Prerequisites}\dotfill \pageref*{prerequisites} \linebreak
\noindent\hyperlink{representations_and_associated_bundles}{$\infty$-Representations and associated $\infty$-bundles}\dotfill \pageref*{representations_and_associated_bundles} \linebreak
\noindent\hyperlink{homotopy_stabilizer_groups}{Homotopy stabilizer groups}\dotfill \pageref*{homotopy_stabilizer_groups} \linebreak
\noindent\hyperlink{KSExtensions}{Higher Kostant-Souriau extensions}\dotfill \pageref*{KSExtensions} \linebreak
\noindent\hyperlink{DefiniteForms}{Definite forms}\dotfill \pageref*{DefiniteForms} \linebreak
\noindent\hyperlink{bps_charges}{BPS Charges}\dotfill \pageref*{bps_charges} \linebreak
\noindent\hyperlink{for_a_single_brane_species}{For a single $p$-brane species}\dotfill \pageref*{for_a_single_brane_species} \linebreak
\noindent\hyperlink{for_branes_ending_on_branes}{For $p_1$-branes ending on $p_2$-branes}\dotfill \pageref*{for_branes_ending_on_branes} \linebreak
\noindent\hyperlink{M5BraneChargesInAnM2BraneCondensate}{Example: M5-brane charges in an M2-brane condensate}\dotfill \pageref*{M5BraneChargesInAnM2BraneCondensate} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{motivation_and_results}{}\subsection*{{Motivation and results}}\label{motivation_and_results}

Consider $(X,g)$ a [[super-spacetime]] and $\omega$ a degree-$(p+2)$ [[differential form]] on $X$ which is a [[geometry of physics -- WZW terms|WZW]] [[curvature form]] [[definite form|definite]] on the [[super Lie algebra]] [[Lie algebra cohomology|cocycle]]

\begin{displaymath}
\overline \psi \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p}
  \in 
  CE(\mathbb{R}^{d-1,1\vert N})
\end{displaymath}
for [[Green-Schwarz super p-brane sigma model]] with [[target space]] $X$. Then the \href{super+Poincare+Lie+algebra#PolyvectorExtensions}{Polyvector extensions}

\begin{displaymath}
[Q_\alpha, Q_\beta] = (\Gamma^a C)_{\alpha\beta} P_a + (\Gamma^{a_1\cdots a_p}C)_{\alpha \beta} Z_{a_1\cdots a_p}
\end{displaymath}
of the [[super Lie algebra]] of [[super-isometries]] of $(X,g)$ by [[charges]] $Z$ of [[Noether currents]] of the [[super p-brane sigma model]] are known as algebras of \emph{[[BPS charges]]}. (The spacetime $(X,g,\omega)$ is called a \emph{[[supergravity]] $\frac{1}{k}$-[[BPS state]]} if the dimension of the space of [[supercharges]] $Q$ in the [[kernel]] of the above bracket is $\frac{1}{k}$th of that of [[super Minkowski spacetime]]).

This is well understood in the literature (\hyperlink{AGIT89}{Azc\'a{}rraga-Gauntlett-Izquierdo-Townsend 89}) for the case that $X$ is locally modeled on an ordinary [[super Minkowski spacetime]] and that the $p$-brane species is in the \emph{old} [[brane scan]] (e.g the [[type II superstrings]], the [[heterotic superstring]] and the [[M2-brane]], also e.g. the [[super 1-brane in 3d]] and the [[3-brane in 6d]], but not the [[D-branes]] and not the [[M5-brane]]).

For the full story of [[string theory]] this needs to be refined in three ways (see [[schreiber:The brane bouquet|Fiorenza-Sati-Schreiber 13]]), and this has been left open in the literature, for previous lack of a [[higher differential geometry]] that could handle this:

\begin{enumerate}%
\item For a genuine global definition of the [[Green-Schwarz super p-brane sigma model]] with target $(X,\omega)$, the WZW [[curvature]] form $\omega$ needs to be [[prequantization|prequantized]] to a globally well-defined [[geometry of physics -- WZW terms|WZW term]], a genuine [[cocycle]] in [[Deligne cohomology]] (a [[circle n-bundle with connection|circle (p1+1)-bundle with connection]]).

(The need for this has broadly been ignored, one place where it is mentioned is (\hyperlink{Witten86}{Witten 86, p. 17}).)


\item For the inclusion of charges of $p_2$-branes on which $p_1$-branes may end (for $p_1 = 1$: [[type II strings]] ending on [[D-branes]], for $p_1= 2$ and $p_2 = 5$ [[M2-branes]] ending on [[M5-branes]]) then $X$ is to be locally modeled on an [[extended super Minkowski spacetime]], hence on a [[super formal smooth infinity-groupoid|super orbispace]], hence a curved spacetime now is an object in [[higher Cartan geometry]] and one needs to make sense of [[Noether currents]] there.

(Arguments in this direction for the [[D-branes]] have been given in (\href{BPS+state#Hammer97}{Hammer 97}) and for the [[M5-brane]] in (\href{BPS+state#SorokinTownsend97}{Sorokin-Townsend 97}).)


\item For inclusion of non-infinitesimal isometries one needs the global structure of the full [[supergroup]] of BPS charges, not just its [[super Lie algebra]].



\end{enumerate}
Here we discuss how to solve these problems in full generality (\hyperlink{SatiSchreiber15}{Sati-Schreiber 15}). Specified to the situation in [[11-dimensional supergravity]] with [[M2-branes]] and [[M5-branes]] we find that the BPS charges traditionally seen in the [[M-theory super Lie algebra]] as living in [[ordinary cohomology]] $H^2(X) \oplus H^5(X)$ of spacetime $X$ receive corrections by $d_4$-differentials of a [[Serre spectral sequence]] given by [[cup product]] with the class of the [[supergravity C-field]]. This is in higher analogy to how [[D-brane charges]] are well known (\href{D-brane#MaldacenaMooreSeiberg01}{Maldacena-Moore-Seiberg 01}) to be in [[ordinary cohomology]] only up to corrections of the $d_3$-differential (and higher) in an [[Atiyah-Hirzebruch spectral sequence]] for [[twisted K-theory]], given by [[cup product]] with the class of the [[B-field]]. This supports the conjecture (\href{M5-brane+charge#Sati10}{Sati 10}) that [[M5-brane charge]] should really be in [[twisted generalized cohomology|twisted]] [[elliptic cohomology]], since this is what is canonically twisted by these degree-4 classes (\href{tmf#ABG10}{Ando-Blumberg-Gepner 10}). (Realizing this fully amounts to refining the term $\mathbf{L}_{M5}^X$ that we construct in [[ordinary differential cohomology]] \hyperlink{M5BraneChargesInAnM2BraneCondensate}{below} to ellitptic differential cohomology. Discussion of that refinement is beyond the scope of this page here.)

We also close a gap in (\hyperlink{AGIT89}{AGIT89}): what is strictly derived there from the [[Noether theorem]] is extension of the [[supersymmetry]] algebra by [[differential forms]], while the argument that it is only the [[de Rham cohomology]] class of these forms that matters relies on physics intuition. We find here that the [[Lie algebra]] of [[conserved currents]] extending the (super-)isometry algebra is naturally not just a ([[super Lie algebra|super-]])[[Lie algebra]] but a ([[super L-infinity algebra|super-]])[[Lie n-algebra|Lie (p+1)-algebra]] including higher order symmetries of Noether symmetries. It is by quotienting these out when restricting the current Lie $n$-algebra to its lowest [[Postnikov tower|Postnikov stage]] that current forms pass to their de Rham equivalence classes. Accordingly, the fully globalized current groups that we find are really ([[super smooth infinity-groupoid|super-]])[[smooth infinity-group|smooth n-groups]]. For instance the [[M-theory super Lie algebra]] is refines to a super Lie 6-group, where $6 = 5+1$ is the dimension of the [[M5-brane]] [[worldvolume]].

$\,$

We formulate the theory general abstractly in the context of [[higher differential geometry]] given by an [[(∞,1)-topos]] $\mathbf{H}$ equipped with [[cohesion]] and [[differential cohesion]]. The application to [[supergravity]] takes place in the model $\mathbf{H} =$ [[SuperFormalSmooth∞Grpd]].

\hypertarget{prerequisites}{}\subsubsection*{{Prerequisites}}\label{prerequisites}

For ease of reference, we recall here some definitions and propositions form previous chapters of \emph{[[geometry of physics]]} which we need for the discussion of BPS charge groups below.

\hypertarget{representations_and_associated_bundles}{}\paragraph*{{$\infty$-Representations and associated $\infty$-bundles}}\label{representations_and_associated_bundles}

\begin{quote}%
recalled from \emph{[[geometry of physics -- representations and associated bundles]]}


\end{quote}
\begin{defn}
\label{FiberBundle}\hypertarget{FiberBundle}{}
Given $V,E\in \mathbf{H}$, a \emph{$V$-[[fiber ∞-bundle]]} $E$ over $X$ is a [[bundle]] $E \in \mathbf{H}_{/X}$ such that there exists a [[cover]] (i.e. a [[1-epimorphism]]) $U \longrightarrow X$ and a [[homotopy pullback]] [[diagram]] of the form

\begin{displaymath}
\itexarray{
    U \times V &\longrightarrow& E
    \\
    \downarrow && \downarrow
    \\
    U &\longrightarrow& X
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{AssociatedBundles}\hypertarget{AssociatedBundles}{}
Given an [[∞-group]] $G$ and a $G$-[[∞-action]] on $V$, and given an $G$-[[principal ∞-bundle]] $P \in \mathbf{H}_{/X}$ [[modulating morphism|modulated]] by $\mathbf{c} \colon X \longrightarrow \mathbf{B}G$, then the \emph{[[associated ∞-bundle]]} is $V$-[[fiber ∞-bundle]] $E = P \times_G V$ which is the [[homotopy pullback]] in

\begin{displaymath}
\itexarray{
     P \times_G V &\longrightarrow& V/G
     \\
     \downarrow && \downarrow
     \\
     X &\stackrel{\mathbf{c}}{\longrightarrow}& \mathbf{B}G
  }
  \,.
\end{displaymath}
A $V$-fiber bundle realized this way is said to have \emph{structure group} $G$.

\end{defn}
\begin{prop}
\label{FiberBundlesAreAssociated}\hypertarget{FiberBundlesAreAssociated}{}
Every $V$-[[fiber ∞-bundle]] is the [[associated ∞-bundle]], def. \ref{AssociatedBundles}, of some $\mathbf{Aut}(V)$-[[principal ∞-bundle]].

\end{prop}
\begin{defn}
\label{ReductionLiftOfStructureGroup}\hypertarget{ReductionLiftOfStructureGroup}{}
Given a $G$-[[principal ∞-bundle]] $P$ [[modulating morphism|modulated]] by some $\mathbf{c}\colon X \longrightarrow \mathbf{B}G$, and given a [[homomorphism]] of [[∞-groups]] $H \hookrightarrow$, then a \emph{[[reduction and lift of structure groups|reduction/lift of the structure group]]} is a lift $\hat {\mathbf{c}}$ in

\begin{displaymath}
\itexarray{
    && \mathbf{B}G
    \\
    &{}^{\mathllap{\hat{\mathbf{c}}}}\nearrow& \downarrow
    \\
    X &\stackrel{\simeq}{\longrightarrow}& \mathbf{B}G
  }
  \,.
\end{displaymath}
Similarly for $V$-[[fiber ∞-bundles]] via def. \ref{AssociatedBundles}, prop. \ref{AssociatedBundles}.

\end{defn}
\hypertarget{homotopy_stabilizer_groups}{}\paragraph*{{Homotopy stabilizer groups}}\label{homotopy_stabilizer_groups}

\begin{quote}%
recalled from \emph{[[geometry of physics -- representations and associated bundles]]}


\end{quote}
For $\mathbf{H}$ an [[(∞,1)-topos]], $G\in \mathbf{H}$ an object equipped with [[∞-group]] structure, hence with a [[delooping]] $\mathbf{B}$G, and for $\rho$ an [[∞-action]] of $G$ on some $V$, exhibited by a [[homotopy fiber sequence]] of the form

\begin{displaymath}
\itexarray{
    V &\stackrel{i}{\longrightarrow}& V/G
    \\
    && \downarrow^{\mathrlap{p_\rho}}
    \\
    && \mathbf{B}G
  }
  \,.
\end{displaymath}
\begin{defn}
\label{StabilizerInInfinityTopos}\hypertarget{StabilizerInInfinityTopos}{}
Given a [[global element]] of $V$

\begin{displaymath}
x \colon \ast \to X
\end{displaymath}
then the \textbf{stabilizer $\infty$-group} $Stab_\rho(x)$ of the $G$-action at $x$ is the [[loop space object]]

\begin{displaymath}
Stab_\rho(x) \coloneqq \Omega_{i(x)} (X/G)
 \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{StabilizerGroupAsFactorization}\hypertarget{StabilizerGroupAsFactorization}{}
Equivalently, def. \ref{StabilizerInInfinityTopos}, gives the [[loop space object]] of the [[1-image]] $\mathbf{B}Stab_\rho(x)$ of the morphism

\begin{displaymath}
\ast \stackrel{x}{\to} X \to X/G
  \,.
\end{displaymath}
As such the [[delooping]] of the stabilizer $\infty$-group sits in a [[1-epimorphism]]/[[1-monomorphism]] factorization $\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X/G$ which combines with the homotopy fiber sequence of prop. \ref{InfinityAction} to a diagram of the form

\begin{displaymath}
\itexarray{
    \ast &\stackrel{x}{\longrightarrow}& X &\stackrel{}{\longrightarrow}& X/G
    \\
    \downarrow^{\mathrlap{epi}} && & \nearrow_{\mathrlap{mono}} & \downarrow
    \\
    \mathbf{B} Stab_\rho(x)
    &=&
    \mathbf{B} Stab_\rho(x)
    &\longrightarrow&
    \mathbf{B}G
  }
 \,.
\end{displaymath}
In particular there is hence a canonical homomorphism of $\infty$-groups

\begin{displaymath}
Stab_\rho(x) \longrightarrow G
  \,.
\end{displaymath}
However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer $Stab_\rho(x)$ is not a \emph{sub}-group of $G$ in general.

\end{defn}
\hypertarget{KSExtensions}{}\paragraph*{{Higher Kostant-Souriau extensions}}\label{KSExtensions}

\begin{quote}%
recalled from \emph{[[geometry of physics -- prequantum geometry]]}, following (\hyperlink{FRS13a}{FRS 13a})


\end{quote}
Throughout, let $\mathbb{G} \in Grp(\mathbf{H})$ be a [[braided ∞-group]] equipped with a [[Hodge filtration]]. Write $\mathbf{B}\mathbb{G}_{conn}\in$ for the corresponding [[moduli stack]] of [[differential cohomology]].

\begin{example}
\label{StandardChoiceOfBGconn}\hypertarget{StandardChoiceOfBGconn}{}
For $\mathbf{H} =$ [[Smooth∞Grpd]] we have $\mathbb{G} = \mathbf{B}^p (\mathbb{R}/\Gamma)$ for $\Gamma = \mathbb{Z}$ is the [[circle n-group|circle (p+1)-group]]. Equipped with its standard Hodge filtration this gives $\mathbf{B}\mathbb{G}_{conn} = \mathbf{B}^p U(1)_{conn}$ presented via the [[Dold-Kan correspondence]] by the [[Deligne complex]] in degree $(p+2)$.

\end{example}
\begin{defn}
\label{DifferentialConcretification}\hypertarget{DifferentialConcretification}{}
For $X \in \mathbf{H}$, for write

\begin{displaymath}
conc \colon [X,\mathbf{B}\mathbb{G}_{conn}] \longrightarrow \mathbb{G}\mathbf{Conn}(X)
\end{displaymath}
for the [[differential concretification]] of the [[internal hom]].

\end{defn}
This is the proper [[moduli stack]] of $\mathbb{G}$-[[principal ∞-connections]] on $X$ in that a family $U \longrightarrow \mathbb{G}\mathbf{Conn}(X)$ is a \emph{vertical} $\mathbb{G}$-principal $\infty$-connection on $U \times X\to U$.

\begin{example}
\label{}\hypertarget{}{}
For $\mathbf{H} =$[[Smooth∞Grpd]] or =[[FormalSmooth∞Grpd]], for $\mathbb{G} = \mathbf{B}^p U(1)$ the [[circle n-group|circle (p+1)-group]] with its standard Hodge filtration as in example \ref{StandardChoiceOfBGconn}, then for $X$ any [[smooth manifold]] or [[formal smooth manifold]], $(\mathbf{B}^p U(1))\mathbf{Conn}(X)$ is presented via the [[Dold-Kan correspondence]] by the sheaf $U \mapsto Ch_\bullet$ of [[vertical differential form|vertical]] [[Deligne complexes]] on $U \times X$ over $U$.

\end{example}
\begin{defn}
\label{}\hypertarget{}{}
For $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then the [[loop space object]] of the moduli stack of $\mathbb{G}$-principal $\infty$-connections on $X$ is the moduli stack of [[flat ∞-connections]] with gauge group $\Omega \mathbb{G}$

\begin{displaymath}
\Omega_0 (\mathbb{G}\mathbf{Conn}(X))
  \simeq
  (\Omega\mathbb{G})\mathbf{FlatConn}(X)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{PrecompositionActionOnGConn}\hypertarget{PrecompositionActionOnGConn}{}
The canonical [[precomposition action|precomposition]] [[∞-action]] of the [[automorphism ∞-group]] $\mathbf{Aut}(X)$ on $[X,\mathbf{B}\mathbb{G}_{conn}]$ passes along $conc$ to an [[∞-action]] on $\mathbb{G}\mathbf{Conn}(X)$.

\end{prop}
\begin{defn}
\label{QuantomorphismGroup}\hypertarget{QuantomorphismGroup}{}
Given a $\mathbb{G}$-[[principal ∞-connection]] $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ there are the following concepts in [[schreiber:Higher geometric prequantum theory|higher geometric prequantum theory]].

\begin{enumerate}%
\item The \textbf{[[quantomorphism ∞-group]]} is the [[stabilizer ∞-group]] of $\nabla \in \mathbb{G}\mathbf{Conn}(X)$, def. \ref{DifferentialConcretification}, under the $\mathbf{Aut}(X)$-action of \ref{PrecompositionActionOnGConn};

\begin{displaymath}
\mathbf{QuantMorph}(X,\nabla)
  \coloneqq
  \mathbf{Stab}_{\mathbf{Aut}(X)}(conc(\nabla))
  \,.
\end{displaymath}

\item The \textbf{[[Hamiltonian symplectomorphism ∞-group]]}

\begin{displaymath}
\mathbf{HamSymp}(X,\nabla) \longrightarrow \mathbf{Aut}(X)
\end{displaymath}
is the [[1-image]] of the canonical morphism $\mathbf{QuantMorph}(X,\nabla) \longrightarrow \mathbf{Aut}(X)$ from remark \ref{StabilizerGroupAsFactorization}.


\item A \textbf{[[Hamiltonian action]]} of an [[∞-group]] $G$ on $(X,\nabla)$ is an [[∞-group]] homomorphism

\begin{displaymath}
\rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla)
   \;
\end{displaymath}

\item An \textbf{[[∞-moment map]]} is an $\infty$-group homomorphism

\begin{displaymath}
G \longrightarrow \mathbf{QuantMorph}(X,\nabla)
\end{displaymath}

\item The \textbf{[[Heisenberg ∞-group]]} for a given Hamiltonian $G$-action $\rho$ is the [[homotopy pullback]]

\begin{displaymath}
\mathbf{Heis}_G(X,\nabla) \coloneqq \rho^\ast \mathbf{QuantMorph}(X,\nabla)
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
For $\mathbf{H} =$ [[Smooth∞Grpd]], for $X \in SmoothMfd \hookrightarrow \mathbf{H}$ a [[smooth manifold]] and for $\nabla$ a [[prequantum line bundle]] on $X$, then $\mathbf{QuantMorph}(X,\nabla)$ is Soriau's [[quantomorphism group]] covering the [[Hamiltonian diffeomorphism]] group. In the case that $(X, F_\nabla)$ is a [[symplectic vector space]] $X = V$ regarded as a linear symplectic manifold with Hamiltonian action on itself by translation, then $\mathbf{Heis}_{V}(X,\nabla)$ is the traditional [[Heisenberg group]].

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
Since $\mathbf{HamSymp}(X,\nabla)\hookrightarrow \mathbf{Aut}(X)$ is by construction a [[1-monomorphism]], given any $G$-action $\rho \colon G \longrightarrow \mathbf{Aut}(X)$ on $X$, not necessarily Hamiltonian, then the homotopy pullback $\rho^\ast \mathbf{QuantMorph}(X,\nabla)$ is the Heisenberg ∞-group of the maximal sub-$\infty$-group of $G$ which does act via Hamiltonian symplectomorphisms. Therefore we will also write $\mathbf{Heis}_G(X,\nabla)$ in this case.

\end{remark}
The following is the refinement of the [[Kostant-Souriau extension]] to [[higher differential geometry]]

\begin{prop}
\label{TheQuantomorphismGroupExtension}\hypertarget{TheQuantomorphismGroupExtension}{}
Given a $\mathbb{G}$-[[principal ∞-connection]] $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$, there is a [[homotopy fiber sequence]] of the form

\begin{enumerate}%
\item if $\mathbb{G}$ is [[0-truncated]] then

\begin{displaymath}
\itexarray{
     \mathbb{G}\mathbf{ConstFunct}(X)
     &\longrightarrow&
     \mathbf{QuantMorph}(X,\nabla)
     \\
     && \downarrow
     \\
     && \mathbf{HamSymp}(X,\nabla)
     &\stackrel{\mathbf{KS}}{\longrightarrow}&
     \mathbf{B} (\mathbb{G}\mathbf{ConstFunct}(X))
 }
\end{displaymath}

\item if $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then

\begin{displaymath}
\itexarray{
     (\Omega \mathbb{G})\mathbf{FlatConn}(X)
     &\longrightarrow&
     \mathbf{QuantMorph}(X,\nabla)
     \\
     && \downarrow
     \\
     && \mathbf{HamSymp}(X,\nabla)
     &\stackrel{\mathbf{KS}}{\longrightarrow}&
     \mathbf{B} ((\Omega \mathbb{G})\mathbf{FlatConn}(X))
 }
\end{displaymath}


\end{enumerate}
exhibiting the [[quantomorphism ∞-group]] as an [[∞-group extension]] of the [[Hamiltonian symplectomorphism ∞-group]] by the [[moduli stack]] of $\Omega \mathbb{G}-$[[flat ∞-connections]], classified by a [[cocycle]] $\mathbf{KS}$.

\end{prop}
(\hyperlink{FRS13a}{FRS13a})

\begin{example}
\label{}\hypertarget{}{}
In $\mathbf{H} =$ [[Smooth∞Grpd]], let $\mathbb{G} = \mathbf{B}^p U(1)$ be the [[circle n-group|circle (p+1)-group]] and let $X \in SmoothMfd \hookrightarrow Smooth \infty Grpd$ be [[n-connected topological space|p-connected]], then $(\Omega\mathbf{B}^p U(1))\mathbf{FlatConn}(X)\simeq \mathbf{B}^{p}U(1)$. Hence here prop. \ref{TheQuantomorphismGroupExtension} gives

\begin{displaymath}
\itexarray{
        \mathbf{B}^{p}U(1)
        &\longrightarrow&
        \mathbf{QuantMorph}(X,\nabla)
        \\
        && \downarrow
        \\
        && \mathbf{HamSymp}(X,\nabla)
        &\stackrel{\mathbf{KS}}{\longrightarrow}&
        \mathbf{B}^{p+1}U(1)
    }
\end{displaymath}
\end{example}
\begin{cor}
\label{KSExtensionForHeis}\hypertarget{KSExtensionForHeis}{}
Given a $\mathbb{G}$-[[principal ∞-connection]] $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$, and for $\rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla)$ a $G$-[[Hamiltonian action]], then there is a [[homotopy fiber sequence]]

\begin{enumerate}%
\item if $\mathbb{G}$ is [[0-truncated]] then

\begin{displaymath}
\itexarray{
     \mathbb{G}\mathbf{ConstFunct}(X)
     &\longrightarrow&
     \mathbf{Heis}_G(X,\nabla)
     \\
     && \downarrow
     \\
     && G
     &\stackrel{\mathbf{KS}(\rho)}{\longrightarrow}&
     \mathbf{B} (\mathbb{G}\mathbf{ConstFunct}(X))
 }
\end{displaymath}

\item if $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then

\begin{displaymath}
\itexarray{
     (\Omega \mathbb{G})\mathbf{FlatConn}(X)
     &\longrightarrow&
     \mathbf{Heis}_G(X,\nabla)
     \\
     && \downarrow
     \\
     && G
     &\stackrel{\mathbf{KS}(\rho)}{\longrightarrow}&
     \mathbf{B} ((\Omega \mathbb{G})\mathbf{FlatConn}(X))
 }
\end{displaymath}


\end{enumerate}
exhibiting the [[Heisenberg ∞-group]] as an [[∞-group extension]] of the $G$ by the [[moduli stack]] of $\Omega \mathbb{G}-$[[flat ∞-connections]], classified by a [[cocycle]] $\mathbf{KS}(\rho)$.

The class of the [[cocycle]] $\mathbf{KS}(\rho)$ is the [[obstruction]] to prequantizing $\rho$ to a [[moment map]] (the \emph{[[classical anomaly]]} of $\rho$); and the the [[Heisenberg ∞-group]] [[∞-group extension|extension]] of $G$ is the universal cancellation of this anomaly.

\end{cor}
\hypertarget{DefiniteForms}{}\subsubsection*{{Definite forms}}\label{DefiniteForms}

The concept of extending a closed [[differential form]] defined on a [[Cartesian space]] $\mathbb{R}^n$ to a \emph{[[definite form]]} on an $n$-dimensional manifold is familiar from [[special holonomy]] manifolds. For instance a definite globalization of the [[associative 3-form]] on $\mathbb{R}^7$ to a 7-manifold induces and is induced by a [[G2-structure]]. But by the discussion at \emph{[[geometry of physics -- prequantum geometry]]}, whenever we see a closed differential form we have to ask whether it is the [[curvature]] of a [[cocycle]] in [[differential cohomology]], hence we have to ask for a higher [[prequantization]]. Here we consider the concept of [[definite forms]] prequantized to such \emph{[[definite globalizations of WZW terms]]}.

\begin{prop}
\label{FunctionsOnTotalSpacesAreSectionsOfFunctionBundle}\hypertarget{FunctionsOnTotalSpacesAreSectionsOfFunctionBundle}{}
Given a $V$-[[fiber ∞-bundle]] $E$ over $X$, def. \ref{FiberBundle}, and given any [[coefficient]] $A$, there is a [[natural equivalence]] beween

\begin{itemize}%
\item morphisms $E \longrightarrow A$;


\item [[sections]] of the canonically [[associated ∞-bundle]] $P \times_{\mathbf{Aut}(V)} [V,A]$ over $X$.



\end{itemize}
\end{prop}
\begin{defn}
\label{DefiniteSection}\hypertarget{DefiniteSection}{}
Given a $V$-[[nLab:fiber ∞-bundle]] $E$ over $X$, and a [[nLab:global element]] $x\colon \ast \to V$ then a [[nLab:section]] $\sigma$ of $E$ is \emph{definite on $x$ if there exists a [[nLab:1-epimorphism]] $U \to X$ and a diagram}

\begin{displaymath}
\itexarray{
    U &\longrightarrow& \ast 
    \\
    \downarrow & \swArrow & \downarrow^{\mathrlap{x}}
    \\
    X & \stackrel{\sigma}{\longrightarrow} & V/\mathbf{Aut}(V)
    \\
    & \searrow & \downarrow
    \\
    && \mathbf{B}\mathbf{Aut}(V)
  }
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{DefiniteSectionsAndReductionOfStructureGroup}\hypertarget{DefiniteSectionsAndReductionOfStructureGroup}{}
Choices of sections definite on $x$ are equivalent to [[reduction of structure groups|reductions of the structure group]], def. \ref{ReductionLiftOfStructureGroup}, along the [[stabilizer group]] map $Stab_\mathbf{Aut(V)}(x)\longrightarrow \mathbf{Aut}(V)$.

\end{prop}
\begin{defn}
\label{DefiniteParameterization}\hypertarget{DefiniteParameterization}{}
Given $\mathbf{c} \colon V \longrightarrow A$, and given a $V$-[[fiber ∞-bundle]] $E$ over $X$, then a \emph{definite parameterization of $\mathbf{c}$ over $E$} is a $\mathbf{c}^E \colon E \longrightarrow A$ such that the section $\sigma_{\mathbf{c}^X}$ coresponding to it via prop. \ref{FunctionsOnTotalSpacesAreSectionsOfFunctionBundle} is definite on $\mathbf{c}$ in the sense of def. \ref{DefiniteSection}.

\end{defn}
\begin{defn}
\label{DefiniteGlobalization}\hypertarget{DefiniteGlobalization}{}
For $V$ an [[∞-group]], $\mathbf{L}\colon V \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ a [[geometry of physics -- WZW terms|WZW term]], and for $X$ a [[geometry of physics -- manifolds and orbifolds|V-manifold]], then a \emph{[[definite globalization of WZW terms|definite globalization]]} of $\mathbf{L}$ over $X$ is

\begin{enumerate}%
\item A [[reduction of the structure group]], def. \ref{ReductionLiftOfStructureGroup}, of the [[frame bundle]] of $X$ along

\end{enumerate}
\begin{displaymath}
\mathbf{Aut}_{Grp}(\mathbb{D}^V) = \mathbf{Aut}^{\ast/}(\mathbf{B}\mathbb{D}^V) \longrightarrow \mathbf{Aut}(\mathbb{D}^V) = GL(V)
    \,,
\end{displaymath}
for $\mathbb{D}^V$ [[generalized the|the]] [[infinitesimal disk]] in $V$;

\begin{enumerate}%
\item an $\mathbf{L}^X \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ such that its pullback $T_{inf} X \to X \stackrel{\mathbf{L}^X}{\longrightarrow} \mathbf{B} \mathbb{G}_{conn}$ to the [[infinitesimal disk bundle]] of $X$ is definite, def. \ref{DefiniteParameterization}, on $\mathbf{L}|_{\mathbb{D}^V} \in \mathbb{G}\mathbf{Conn}(\mathbb{D}^V)$, def. \ref{DifferentialConcretification}.

\end{enumerate}
Since, according to prop. \ref{DefiniteSectionsAndReductionOfStructureGroup}, the second item in def. \ref{DefiniteGlobalization} implies a [[reduction of the structure group|lift/reduction of the structure group]] to $\mathbf{QuantMorph}(\mathbf{L}|_{\mathbb{D}^V})$, in total this requires a reduction/lift to the [[Heisenberg ∞-group]]

\begin{displaymath}
G =
  \mathbf{Heis}_{\mathbf{Aut}_{grp}(\mathbb{D}^V)}(\mathbf{L}|_{\mathbb{D}^V})
  \coloneqq
  \mathbf{QuantMorp}(\mathbf{L}|_{\mathbb{D}^V})
  \underset{\mathbf{Aut}(\mathbb{D}^V)}{\times}
  \mathbf{Aut}_{Grp}(\mathbb{D}^V)
 \,.
\end{displaymath}
This [[G-structure]] we require to be [[integrable G-structure|first order integrable]] (with respect to the canonical left-invariant [[framed manifold|framing]] of $V$.)

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
Let $\mathbf{H} =$ [[Smooth∞Grpd]].

\begin{enumerate}%
\item For $\mathbf{L}\colon T^\ast \mathbb{R}^n \to \mathbf{B}U(1)_{conn}$ the [[Liouville-Poincaré 1-form]] $\theta = \sum_{i = 1}^n p_i d q^i$ (regarded as a [[principal connection]] on the trivial [[circle bundle]]), then a definite globalization of $\mathbf{L}$ is a [[symplectic manifold]] equipped with a [[prequantum line bundle]].


\item for $\mathbf{L}\colon \mathbb{R}^7 \longrightarrow \mathbf{B}^3 U(1)_{conn}$ a potential for the [[associative 3-form]], then a definite globalization is a manifold with [[G2-structure]] $(X,\omega_3)$ equipped with a [[bundle gerbe with connection]] whose 3-form curvature is $\omega_3$.



\end{enumerate}
\end{example}
\begin{example}
\label{11dSugreEquationsOfMotionFromDefiniteGlobalization}\hypertarget{11dSugreEquationsOfMotionFromDefiniteGlobalization}{}
In $\mathbf{H} =$ [[SuperFormalSmooth∞Grpd]] and for $V$ being [[super Minkowski spacetime]] of bosonic [[dimension]] $d = 3,4,10,11$ regarded as the [[supersymmetry]] [[super-translation group]] in that dimension, and for $\mathbf{L} = \mathbf{L}$ the WZW term induced by differential Lie integration (\href{geometry+of+physics+--+WZW+terms#WZWTermFromLieIntegration}{here}) from the [[super Lie algebra]] [[Lie algebra cohomology|cocycles]] of the [[brane scan]] in these dimensions, then the [[Heisenberg ∞-group]] in def. \ref{DefiniteGlobalization} is a $\mathbf{B}(\mathbb{R}/\Gamma)$-[[∞-group extension]] of the [[Lorentz group]] in these dimensions.

This means that a choice of definite globalization of $\mathbf{L}_{string}$ over a [[supermanifold]] $X$ is in particular a choice of super-[[orthogonal structure]], hence a choice of [[graviton]] and of a [[gravitino]] [[field (physics)|field]].

The condition that this [[G-structure]] be first-order integrable with respect to the canonical left-invariant framing of [[super Minkowski spacetime]] then means that the [[supertorsion]] of this orthogonal structure vanishes.

For $d = 1$ this is the [[torsion constraint of supergravity]]. By (\href{torsion+constraints+in+supergravity#CandielloLechner93}{Candiello-Lechner 93}, \href{torsion+constraints+in+supergravity#Howe97}{Howe 97}) this implies that the above graviton and gravitino field satisfy the [[Einstein equations]] for bosonic backgrounds of [[11-dimensional supergravity]].

This in turn implies in particular that the [[curvature]] of the WZW term $\mathbf{L}$ is the fermionic component of the [[supergravity C-field]] [[field strength]]. This finally means that $\mathbf{L}$ itself is a consistent choice of [[prequantization]] of this hence a genuinely globally defined WZW term for the [[Green-Schwarz sigma model]] for the [[M2-brane]] with [[target space]] $X$.

\end{example}
\hypertarget{bps_charges}{}\subsubsection*{{BPS Charges}}\label{bps_charges}

Once a $V$-manifold $X$ is equipped with a definite globalization $\mathbf{L}^X$ of a WZW term $\mathbf{L}$, according to \ref{DefiniteGlobalization}, and hence also with a [[G-structure]] $\mathbf{g}$ for $G$ the suitable [[homotopy stabilizer group]] of $\mathbf{L}$ on [[infinitesimal disks]], then the [[automorphism ∞-group]] $\mathbf{Aut}(X)$ is naturally ``[[spontaneously broken symmetry|broken]]'' to the [[homotopy stabilizer group]] of this extra data. The stabilizer of the $G$-structure itself yields the [[isometry group]] $\mathbf{Iso}(X,\mathbf{g})$, but since the higher WZW term has in general [[higher gauge symmetries]], the total homotopy stabilizer of the triple $(X,\mathbf{g},\mathbf{L}^{X})$ is a [[Heisenberg ∞-group|Heisenberg]] [[∞-group extension]] of that. Since for the case of applications to [[supergravity]] (examples \ref{11dSugreEquationsOfMotionFromDefiniteGlobalization}, \ref{M5ChargeExtensions} below) the [[0-truncated|0-truncation]] of this [[∞-group extension]] turns out to be the extension by [[BPS charges]], we here speak, for lack of any other established term, generally of \emph{BPS charge groups} for homotopy stabilizers of definitely globalized higher WZW terms.

\hypertarget{for_a_single_brane_species}{}\paragraph*{{For a single $p$-brane species}}\label{for_a_single_brane_species}

\begin{defn}
\label{BPSChargeGroup}\hypertarget{BPSChargeGroup}{}
Given a definite globalization $\mathbf{L}^X \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ of a WZW term, def. \ref{DefiniteGlobalization}, hence in particular a [[G-structure]] $\mathbf{g}$ on $X$ for $G =
  \mathbf{Heis}_{\mathbf{Aut}_{grp}(\mathbb{D}^V)}(\mathbf{L}|_{\mathbb{D}^V})$, then the corresponding \emph{BPS charge group} is the [[Heisenberg n-group]], def. \ref{QuantomorphismGroup}, of $\mathbf{L}^X$ over the [[isometry group]] of this $G$-structure:

\begin{displaymath}
BPS(X,\mathbf{g}, \mathbf{L})
  \coloneqq
  \mathbf{Heis}_{\mathbf{Iso}(X,\mathbf{g})}(X,\mathbf{L}^X)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $V$ [[super Minkowski spacetime]], then the [[super L-∞ algebra]] of $BPS(X,\mathbf{g}, \mathbf{L})$, def. \ref{BPSChargeGroup} is the [[Poisson bracket L-∞ algebra]] $\mathfrak{Pois}(X,\omega)$ of $\omega = F_{\mathbf{L}}$ regarded as a [[pre-n-plectic form|pre-(p+1)-plectic form]] on $X$. See the discussion in (\hyperlink{FRS13b}{FRS 13b, section 4}).

\end{remark}
Accordingly we find $L_\infty$-algebraic versions of the higher Kostant-Heisenberg $\infty$-extensions of prop. \ref{LieKSExtension}:

\begin{prop}
\label{LieKSExtension}\hypertarget{LieKSExtension}{}
There is a [[homotopy fiber sequence]] in the [[homotopy theory of L-∞ algebras]] of the form

\begin{displaymath}
\itexarray{
    \mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R})
    &\longrightarrow&
    \mathfrak{Pois}_{Iso(X,g)}(X,\omega)
    \\
    && \downarrow
    \\
    && HamIso(X,g,\omega)
    &\stackrel{ks}{\longrightarrow}&
    \mathbf{B}\mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R})
  }
\end{displaymath}
which exhibits the [[Poisson bracket L-∞ algebra]] as an [[L-∞ algebra extension]] of the Lie algebra of $\omega$-Hamiltonian [[Killing vectors]] and [[Killing spinors]] by the truncated [[de Rham complex]]

\begin{displaymath}
\mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R})
  \simeq
  (\Omega^0(X)\stackrel{d}{\to} \Omega^1(X)\stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^p_{cl}(X))
\end{displaymath}
in degree $p$, regarded as an abelian $L_\infty$-algebra.

\end{prop}
(\hyperlink{FRS13b}{FRS 13b, theorem 3.3.1})

\begin{remark}
\label{}\hypertarget{}{}
By (\hyperlink{FRS13b}{FRS 13b, theorem 4.2.2}) $\mathfrak{Pois}(X,\omega)$ has a model by the [[dg-Lie algebra]] (\hyperlink{FRS13b}{FRS 13b, def./prop. 4.2.1}). Its bracket in degree-0 (\hyperlink{FRS13b}{FRS 13b, equation (4.2.1)}) is the bracket of [[Noether currents]] for $\omega$ regarded as a WZW curvature as considered in (\hyperlink{AGIT89}{AGIT 89}).

\end{remark}
But $\mathfrak{Pois}(X,\omega)$ encodes also the higher order currents between these currents, which get quotiented out when passing to its degree-0 [[Postnikov stage]]:

\begin{prop}
\label{0TruncationOfHigherPoissonBracket}\hypertarget{0TruncationOfHigherPoissonBracket}{}
For connective [[L-∞ algebras]], 0-truncation yields a [[functor]]

\begin{displaymath}
\tau_0
  \colon
  L_\infty Alg_{\geq 0}
  \longrightarrow
  LieAlg
\end{displaymath}
to [[Lie algebras]]. Under this functor this higher Kostant-Soriau extension of prop. \ref{LieKSExtension} becomes a [[Lie algebra extension]]

\begin{displaymath}
0 \to H^p_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega)
  \longrightarrow 
  Ham(X,\omega)
  \to
  0
\end{displaymath}
of the [[Hamiltonian vector fields]] by the degree-$p$ [[de Rham cohomology]] group of $X$, regarded as an abelian Lie algebra.

\end{prop}
\hypertarget{for_branes_ending_on_branes}{}\paragraph*{{For $p_1$-branes ending on $p_2$-branes}}\label{for_branes_ending_on_branes}

Consider now two \href{http://ncatlab.org/nlab/show/geometry+of+physics+--+WZW+terms#ConsecutiveWZWTermsAndTwists}{consecutive WZW terms}

\begin{displaymath}
\itexarray{
    \widetilde{\hat V} &\stackrel{\mathbf{L}_2}{\longrightarrow}&
    \mathbf{B}(\mathbb{G}_2)_{conn}
    \\
    \downarrow
    \\
    V &\stackrel{\mathbf{L}_1}{\longrightarrow}&
    \mathbf{B}(\mathbb{G}_1)_{conn}
  }
\end{displaymath}
with $\mathbf{L}_2$ defined on the differential refinement of the [[∞-group]] extension

\begin{displaymath}
\itexarray{
    \mathbb{G} &\longrightarrow& \hat V
    \\
    && \downarrow
    \\
    && V &\stackrel{}{\longrightarrow}& \mathbf{B}\mathbb{G}
  }
\end{displaymath}
which is the underlying $\mathbb{G}$-[[principal ∞-bundle]] underlying $\mathbf{L}_1$.

\begin{displaymath}
\itexarray{
    \widetilde{\hat V} &\stackrel{}{\longrightarrow}&
    \Omega^1(-,\mathbb{G})
    \\
    \downarrow &(pb)& \downarrow
    \\
    V &\stackrel{\mathbf{L}_1}{\longrightarrow}&
    \mathbf{B}(\mathbb{G}_1)_{conn}
  }
\end{displaymath}
\begin{prop}
\label{ExtendedSpacetimeIsConsistent}\hypertarget{ExtendedSpacetimeIsConsistent}{}
Given two \href{http://ncatlab.org/nlab/show/geometry+of+physics+--+WZW+terms#ConsecutiveWZWTermsAndTwists}{consecutive WZW terms}, $(\mathbf{L}_1,\mathbf{L}_2)$ and given a define globalization, def. \ref{DefiniteGlobalization}, of $\mathbf{L}_1$ over a $V$-manifold $X$ then

\begin{enumerate}%
\item the isometry action canonically lifts from $X$ to to the extended spacetime $\widetilde{\hat X}$

\end{enumerate}
\begin{displaymath}
\itexarray{
       \widetilde {\hat X} &\longrightarrow& \Omega^1(-\mathbb{G})
       \\
       \downarrow &(pb)& \downarrow 
       \\
       X &\stackrel{\mathbf{L}_1}{\longrightarrow}& 
       \mathbf{B}\mathbb{G}_{conn}
     }
     \,;
\end{displaymath}
\begin{enumerate}%
\item the [[infinitesimal disks]] in $\widetilde {\hat X}$ are equivalent to those of $\widetilde {\hat V}$.

\end{enumerate}
\end{prop}
\begin{defn}
\label{BPSGroupForConsecutiveWZWTerms}\hypertarget{BPSGroupForConsecutiveWZWTerms}{}
By this proposition it is consistent to ask for a \emph{consecutive} definitite globalization of two consecutive WZW term

\begin{displaymath}
\itexarray{
    \widetilde{\hat X} &\stackrel{\mathbf{L}_2^X}{\longrightarrow}&
    \mathbf{B}(\mathbb{G}_2)_{conn}
    \\
    \downarrow
    \\
    X &\stackrel{\mathbf{L}_1^X}{\longrightarrow}&
    \mathbf{B}(\mathbb{G}_1)_{conn}
  }
  \,.
\end{displaymath}
The \emph{BPS charge $\infty$-group} of this setup is

\begin{displaymath}
\mathbf{BPS}(X,\mathbf{g}, \mathbf{L}_1^X, \mathbf{L}_2^X)
  \coloneqq
  \mathbf{Heis}_{\mathbf{Iso}(X,\mathbf{g})}(\widetilde {\hat X},\mathbf{L}_2^X)
  \,.
\end{displaymath}
\end{defn}
By cor. \ref{KSExtensionForHeis} this is an [[∞-group extension]] of $\mathbf{Iso}(X,\mathbf{g})$ by $\mathbb{G}_2\mathbf{FlatConn}(\widetilde{\hat X})$. Forgetting the differential part of the twist, this extension group receives a map from $\mathbb{G}_2\mathbf{FlatConn}({\hat X})$.

\hypertarget{M5BraneChargesInAnM2BraneCondensate}{}\subsubsection*{{Example: M5-brane charges in an M2-brane condensate}}\label{M5BraneChargesInAnM2BraneCondensate}

\begin{example}
\label{M5ChargeExtensions}\hypertarget{M5ChargeExtensions}{}
Consider again $\mathbf{H}=$ [[SuperFormalSmooth∞Grpd]] as in example \ref{11dSugreEquationsOfMotionFromDefiniteGlobalization}.

From the [[odd line|super point]] $\mathbb{R}^{0|1} \in \mathbf{H}$ there emanates a [[schreiber:The brane bouquet|bouquet]] of consecutive [[super L-∞ algebra]] [[L-∞ algebra extensions|extensions]], part of which looks as follows (\hyperlink{FSS13}{FSS 13}):

We now concentrate on the branch of this classified by the cocycles $\mu_4$ for the [[M2-brane]], and $\mu_7$ for the [[M5-brane]]:

\begin{displaymath}
\itexarray{
    \mathfrak{m}5\mathfrak{brane}
    \\
    \downarrow
    \\
    \mathfrak{m}2\mathfrak{brane}
    &\stackrel{\mu_7\coloneqq\overline{\psi}\Gamma^{a_1 \cdots a_5}\wedge \psi \wedge e_{a_1}\wedge \cdots \wedge e_{a_5} + c_3\wedge \mu_4}{\longrightarrow}&
    b^6 \mathbb{R}
    \\
    \downarrow
    \\
    \mathbb{R}^{10,1\vert \mathbf{32}}
    &\stackrel{\mu_4 \coloneqq \overline{\psi}\Gamma^{a_1 a_2}\wedge \psi \wedge e_{a_1}\wedge e_{a_2}}{\longrightarrow}&
    b^2 \mathbb{R}
  }
\end{displaymath}
Here $\mathfrak{m}2\mathfrak{brane}$ denotes the ``[[supergravity Lie 3-algebra]]'' regarded as an [[extended super Minkowski spacetime]] and $\mathfrak{m}5\mathfrak{brane}$ denotes the ``[[supergravity Lie 6-algebra]]''. Both hooks $\itexarray{\downarrow \\ & \rightarrow}$ in the diagram are [[homotopy fiber sequences]] in the [[homotopy theory of L-∞ algebras|homotopy theory of super L-∞ algebras]].

By the discussion at \emph{\href{geometry+of+physics+--+WZW+terms#ConsecutiveWZWTermsAndTwists}{geometry of physics -- WZW terms -- Consecutive WZW terms}}, following (\hyperlink{FSS13}{FSS 13}) applying differentially refined [[Lie integration]] to this yields two consecutive higher WZW terms of the form

\begin{displaymath}
\itexarray{
     \mathbf{B}^{2}(\mathbb{R}/\Gamma_1)_{conn} &\longrightarrow& \widetilde {\hat{\mathbb{R}}}^{10,1\vert \mathbf{32}}
     &\stackrel{\mathbf{L}_{M5}}{\longrightarrow}&
     \mathbf{B}^{5} (\mathbb{R}/\Gamma_2)_{conn}
     \\
     && \downarrow
     \\
     && \mathbb{R}^{10,1\vert \mathbf{32}}
     &\stackrel{\mathbf{L}_{M2}}{\longrightarrow}&
     \mathbf{B}^2 (\mathbb{R}/\Gamma_1)_{conn}
  }
  \,.
\end{displaymath}
(Here by the [[van Est isomorphism]] we do not notationally distinguish [[super Minkowski spacetime]] regarded as a [[super Lie algebra]] or a [[super Lie group]].)

By example \ref{11dSugreEquationsOfMotionFromDefiniteGlobalization} a choice of definite globalization of $\mathbf{L}_{M2}$ is equivalent to bosonic solution $(X,\mathbf{g})$ to the [[Einstein equations]] of [[11-dimensional supergravity]] equipped with a compatible globally defined WZW term $\mathbf{L}_{M2}^X$ for the [[M2-brane]] [[Green-Schwarz sigma model]] with [[target space]] $X$.

By prop. \ref{ExtendedSpacetimeIsConsistent} this defines an extended superspacetime $\widetilde{\hat X}$ which is a [[higher Cartan geometry]] of locally modeled on the [[supergravity Lie 3-algebra]] on which the local [[M5-brane]] [[Green-Schwarz sigma model]] is defined, and hence may ask for a choice of definite globalization of that

\begin{displaymath}
\itexarray{
     \widetilde {\hat{X}}
     &\stackrel{\mathbf{L}^X_{M5}}{\longrightarrow}&
     \mathbf{B}^{5} (\mathbb{R}/\Gamma_2)_{conn}
     \\
     \downarrow
     \\
     X
     &\stackrel{\mathbf{L}^X_{M2}}{\longrightarrow}&
     \mathbf{B}^2 (\mathbb{R}/\Gamma_1)_{conn}
  }
  \,.
\end{displaymath}
This is now a globally well defined background for the M5-brane sigma model and def. \ref{BPSGroupForConsecutiveWZWTerms} determines its BPS-char super-6-group. By corollary \ref{KSExtensionForHeis} this is a super 6-group extension of the [[superisometry group]] of $(X,\mathbf{g})$ by $(\mathbf{B}^5 (\mathbb{R}/\Gamma_2))\mathbf{FlatConn}(\widetilde{\hat X})$.

By the discussion at \emph{\href{geometry+of+physics+--+WZW+terms#ConsecutiveWZWTermsAndTwists}{geometry of physics -- WZW terms -- Consecutive WZW terms}} the extended spacetime $\widetilde {\hat X}$ here is such that smooth maps into it

\begin{displaymath}
\Sigma \longrightarrow \widetilde{\hat X}
\end{displaymath}
(which are the [[field (physics)|fields]] of the [[M5-brane]] [[sigma model]] with WZW term $\mathbf{L}^X_{M5}$) are equivalently pairs, consisting of

\begin{enumerate}%
\item a smooth function $\phi \colon \Sigma \longrightarrow X$ into the actual spacetime $X$;


\item a [[cocycle]] $\nabla$ in $\phi$-twisted degree-3 [[Deligne cohomology]] on $\Sigma$, hence a [[B-field|2-form gauge field]] on $\Sigma$, subject to certain compatibility conditions with the function $\phi$.



\end{enumerate}
The first item here is the evident [[sigma model]] field, the second 2-form field is part of the ``tensor multiplet'' on the [[M5-brane]], exhibiting the Green-Schwarz sigma-model for the M5-brane as a \emph{higher} [[gauged WZW model]].

Now to consider the BPS charge group of $\mathbf{L}^X_{M5}$, def. \ref{BPSGroupForConsecutiveWZWTerms}. By corollary \ref{KSExtensionForHeis} this is an [[∞-group extension]] of the [[super-isometry group]] of the 11-dimensional [[super spacetime]] by the moduli stack $(\mathbf{B}^5 U(1))\mathbf{FlatConn}(\widetilde{\hat X})$ of flat 5-form connection on the extended spacetime.

This receives a map $(\mathbf{B}^5 U(1))\mathbf{FlatConn}(\hat X) \longrightarrow (\mathbf{B}^5 U(1))\mathbf{FlatConn}(\widetilde{\hat X})$ from the moduli of 5-form connections of the extended spacetime $\hat X$ (which is $\widetilde{\hat X}$. This consists of the cohomological data without the \emph{differential} cohomologica data in $\mathbf{L}_{M2}^X$): it is the $\mathbf{B}^2 (\mathbb{R}/\Gamma_1)$-[[principal ∞-bundle]] which sits in the [[homotopy fiber sequence]] of the form

\begin{displaymath}
\itexarray{
    \mathbf{B}^2 (\mathbb{R}/\Gamma_1)
    &\longrightarrow&
    \hat X
    \\
    && \downarrow
    \\
    && X &\stackrel{\mathbf{DD}(\mathbf{L}_{M2}^X)}{\longrightarrow}&
    \mathbf{B}^3 (\mathbb{R}/\Gamma_1)
  }
  \,.
\end{displaymath}
Under [[Lie differentiation]] as in prop. \ref{0TruncationOfHigherPoissonBracket} $(\mathbf{B}^5 U(1))\mathbf{FlatConn}(\widetilde{\hat X})$ turns into $(\mathbf{B}^5 \mathbb{R})\mathbf{FlatConn}(\widetilde{\hat X})$ hence into $\mathbf{H}(\hat X, \flat \mathbf{B}^5 \mathbb{R})$. Under the [[adjunction]] between [[shape modality]] $\int$ and [[flat modality]] $\flat$, this is the degree-5 real cohomology of the [[geometric realization]] of $\hat X$. This in turns is a [[Eilenberg-MacLane space|K(Z,3)]]-fibration $\int \hat X \to \int X$ over the underlying bare [[homotopy type]] of spacetime $X$ which is classified by the integral degree-4 class which is the higher [[Dixmier-Douady class]] $DD(\mathbf{L}_{M5}^X)$ of $\mathbf{L}_{M2}^{X}$.

The degree-5 real cohomology of such a fibration is computed by a [[Serre spectral sequence]]. By the discussion at \emph{\href{Eilenberg-Mac+Lane+space#CohomologyOfEMSpaces}{Eilenberg-MacLane space -- cohomology of EM spaces}} only very few entries in this spectral sequence contribute, and the result is the middle cohomology of this sequence

\begin{displaymath}
H^1(X) \stackrel{(0,d_4)}{\longrightarrow} H^2(X) \oplus H^5(X)
  \stackrel{(d_4,0)}{\longrightarrow}
  H^6(X)
\end{displaymath}
where $d_4 \propto (-)\cup DD(\mathbf{L}_{M2}^X)$ is given by taking the [[cup product]] with the class of the M2-WZW term.

This is the group of M2-brane and M5-brane charges with corrections by global effects, in the corrected [[M-theory super Lie algebra]] for the superspacetime $(X,g)$.

\end{example}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[José de Azcárraga]], [[Jerome Gauntlett]], J.M. Izquierdo, [[Paul Townsend]], \emph{Topological Extensions of the Supersymmetry Algebra for Extended Objects}, Phys. Rev. Lett. 63 (1989) 2443 (\href{http://inspirehep.net/record/26393?ln=en}{spire})


\item [[Edward Witten]], \emph{Twistor-like transform in ten dimensions}, Nuclear Physics B266 (1986) (\href{http://inspirehep.net/record/214192?ln=en}{spire})


\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory]]} (\href{http://arxiv.org/abs/1304.0236}{arXiv:1304.0236})


\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]}, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 -- 142 (\href{http://arxiv.org/abs/1304.6292}{arXiv:1304.6292})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]}, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264})


\item [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Lie n-algebras of BPS charges]]} (\href{http://arxiv.org/abs/1507.08692}{arXiv:1507.08692})



\end{itemize}
All details and proofs for the above are in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{https://dl.dropboxusercontent.com/u/12630719/dcct.pdf}{pdf})

\end{itemize}


\end{document}