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\begin{document}

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\section*{fivebrane 6-group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_lie_theory}{}\paragraph*{{Higher Lie theory}}\label{higher_lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{higher_spin_geometry}{}\paragraph*{{Higher spin geometry}}\label{higher_spin_geometry}

[[!include higher spin geometry - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{fivebrane 6-group} $Fivebrane(n)$ is a smooth version of the [[topological space]] that appears in the second step of the [[Whitehead tower]] of the [[orthogonal group]].

It is a lift of this through the  functor $\Pi :$ [[?LieGrpd]] $\to$ [[∞Grpd]].

One step below the fivebrane 6-group in the Whitehead tower is the [[string Lie 2-group]].

For the time being see the discussions at



and the Motivation section at

[[infinity-Chern-Weil theory]]

for more background.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In the [[(∞,1)-topos]] $\mathbf{H} =$ [[?LieGrpd]] we have a smooth refinement of the second fractional [[Pontryagin class]]

\begin{displaymath}
\frac{1}{6}
  \mathbf{p}_2
  :
  \mathbf{B} String(n) 
  \to 
  \mathbf{B}^7 \mathbb{R}/\mathbb{Z}
\end{displaymath}
defined on the [[delooping]] of the [[string Lie 2-group]]. Strictly speaking, we need $n\gt 6$, since for low $n$, $String(n)$ is not 6-connected. See [[orthogonal group\#HomotopyGroups|orthogonal group]] for a table of the relevant homotopy groups.

The [[delooping]] $\mathbf{B}Fivebrane(n)$ of the \textbf{fivebrane 6-group} is the [[principal ∞-bundle]] classified by this in $\mathbf{H}$, that is the [[homotopy fiber]]

\begin{displaymath}
\itexarray{
    \mathbf{B} Fivebrane(n) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}String(n) &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 \mathbb{R}/\mathbb{Z}
  }
  \,.
\end{displaymath}
\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

Along the lines of the description at [[Lie integration]] and [[string 2-group]], in a canonical [[models for infinity-stack (infinity,1)-toposes|model]] for $\mathbf{H}$ the morphism $\frac{1}{6}\mathbf{p}_2$ is given by a morphism out of a [[resolution]] $\mathbf{B}Q$ of $\mathbf{B}String(n)$ that is built in degree $k \leq 7$ from smooth $k$-[[simplices]] in the [[Lie group]] $Spin(n)$. This morphism assigns to a 7-simplex $\phi : \Delta^7_{Diff} \to Spin(n)$ the integral

\begin{displaymath}
\int_{\Delta^7_{Diff}} \phi^* \mu_7 \;\;\in
  \mathbb{R}/\mathbb{Z}
\end{displaymath}
of the degree 7 [[Lie algebra cohomology|Lie algebra cocycle]] $\mu_7$ of the [[special orthogonal Lie algebra]] $\mathfrak{so}(n)$ which is normalized such that its pullback to $String(n)$ (..explain\ldots{}) is the deRham image of the generator in [[integral cohomology]] there.

More in detail, a resolution of $\mathbf{B}String(n)$ is given by the [[coskeleton]]

\begin{displaymath}
\mathbf{cosk}_7
  \left(
  \itexarray{
    Q_7 \subset hom(\Delta^7_{Diff}, G) \times (U(1))^{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4}
    \\
    \downarrow \downarrow 
    \downarrow\downarrow 
    \downarrow \downarrow    
    \downarrow \downarrow    
    \\
    \vdots
    \\
    \downarrow \downarrow \downarrow\downarrow \downarrow \downarrow    
    \\
    Q_4 \subset hom(\Delta^4_{Diff}, G) \times (U(1))^{20}
   \\    
    \downarrow \downarrow \downarrow\downarrow \downarrow
    \\     
    Q_3 \subset hom(\Delta^3_{Diff}, G) \times (U(1))^4
   \\    
    \downarrow \downarrow \downarrow\downarrow
    \\     
    hom(\Delta^2_{Diff}, G) \times U(1)
   \\    
    \downarrow \downarrow \downarrow
    \\     
    hom(\Delta^1_{Diff}, G)
    \\
    \downarrow \downarrow
    \\
    *
  }
  \right)
\end{displaymath}
where the subobjects are those consisting of 3-simplices in $G$ with 2-faces labeled in $U(1)$ such that the integral of $\mu_3$ over the 3-simplex in $\mathbb{R}/\mathbb{Z}$ is the signed product of these labels.

(\ldots{})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\textbf{fivebrane 6-group} $\to$ [[string 2-group]] $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]]

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

The topological [[fivebrane group]] with its interpretation in [[dual heterotic string theory]] was discussed in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{Fivebrane structures} Reviews in mathematical physics, 10 (2009) 1197 ()

\end{itemize}
and the smooth fivebrane 6-group was indicated. The latter is discussed in more detail in section 4.1 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}


\item [[Jesse Wolfson]] says he has shown the existence of a presentation of the $Fivebrane$ [[smooth infinity-group|smooth 6-group]] by a locally Kan and degreewise finite-dimensional simplicial smooth manifold.



\end{itemize}
[[!redirects Fivebrane 6-group]]

[[!redirects fivebrane Lie 6-group]] [[!redirects Fivebrane Lie 6-group]]

[[!redirects smooth fivebrane 6-group]]



\end{document}