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\begin{document}

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\section*{Fivebrane structure}

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

\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{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The notion of \emph{Fivebrane structure} is the next higher analog of that of [[spin structure]] and [[string structure]].

Recall from the discussion there that a [[string structure]] on [[manifold]] $X$ with [[spin structure]] is a lift $\hat g$ of the classifying map $g : X \to B Spin(n)$ of the [[tangent bundle]] associated to a [[Spin group]]-[[principal bundle]] through the next step in the [[Whitehead tower]] of $O(n)$, called $B String(n)$ -- the [[delooping]] of the [[String group]]:

\begin{displaymath}
\itexarray{
    && B String(n)
    \\
    & {\hat g}\nearrow & \downarrow
    \\
    X &\stackrel{g}{\to}& B Spin(n)
  }
  \,.
\end{displaymath}
The names ``Spin'' and ``String'' both derive from the role these structures play in [[quantum field theory]]: a [[spin structure]] is required on $X$ for it to serve as a target space for spinning particles (superparticles), while a [[string structure]] is required for it to serves as a target for ``spinning strings'' -- superstrings -- (see [[heterotic string theory]] for more). Topologists just say (said) $O(n)\langle 2\rangle$ for $Spin(n)$ and $O(n)\langle 6\rangle$ for $String(n)$, respectively.

They wrote $O(n)\langle 8\rangle$ for the next step in the [[Whitehead tower]] of $O(n)$ (note that this is only the \emph{next} step for $n \gt 6$; for lower $n$ there are intermediate steps, as can be seen in the table at \href{orthogonal+group#HomotopyGroups}{orthogonal group}).

It was [[Hisham Sati]] who first realized that a lift of the [[tangent bundle]] $T X$ to this highly connected structure group is related to $X$ serving as a target for ``spinning 5-branes'' -- super-5-branes -- in what is called [[dual heterotic string theory]]. Following the history of the term [[String group]] he gave the topological group $O(n)\langle 8\rangle$ the name [[Fivebrane group]]: $Fivebrane(n)$.

Accordingly, a \textbf{Fivebrane structure(n)} on a manifold $X$ with [[string structure]] is a lift of $\hat g : X \to B String(n)$ to $\hat \hat g$

\begin{displaymath}
\itexarray{
    && B Fivebrane(n)
    \\
    & {\hat \hat g}\nearrow & \downarrow
    \\
    X &\stackrel{\hat g}{\to}& B String(n)
  }
  \,.
\end{displaymath}
The obstruction class to this lift is a fractional multiply of the second [[Pontrjagin class]]. Namely the generator of $H^8(B String, \mathbb{Z})$ is $\frac{1}{6}p_2$,

\begin{displaymath}
\itexarray{
     B String &\stackrel{1/6 p_2}{\to}& B^8 \mathbb{Z}
     \\
     \downarrow && \downarrow^{\mathrlap{\cdot 6}}
     \\
     B SO &\stackrel{p_2}{\to}& B^8 \mathbb{Z}
  }
  \,.
\end{displaymath}
(stated in \hyperlink{SSS2}{SSS2}, then in \hyperlink{DHH}{DHH}, also follows from the [[index theory]] argument leading to (3.3) in \hyperlink{Witten96}{Witten 96}).

The [[Fivebrane group]] is the [[loop space object]] of the corresponding [[homotopy fiber]]

\begin{displaymath}
\itexarray{
    B Fivebrane &\to& * 
    \\
    \downarrow && \downarrow
    \\
    B String &\stackrel{\frac{1}{6}}{\to}& B^7 U(1)&
  }
\end{displaymath}
and so, by the [[universal property]] of the [[homotopy pullback]], String-structures $\hat g$ lift to Fivebrane structures precisely if $\frac{1}{6}p_2(\hat g)$ is trivial in [[cohomology]]

\begin{displaymath}
\itexarray{
    && B Fivebrane &\to& * 
    \\
    & {}^{\mathllap{\hat \hat g}}\nearrow & \downarrow && \downarrow
    \\
    X &\stackrel{\hat g}{\to}& B String &\stackrel{\frac{1}{6}p_2}{\to}& B^7 U(1)
  }
  \,.
\end{displaymath}
In (\hyperlink{SSS2}{SSS2}) the physical interpretation of this topological lift was established by comparison with known [[quantum anomaly]] cancellaton conditions in [[dual heterotic string theory]].

The term ``Fivebrane'' apparently quickly caught on in the mathematical community, for instance in (\hyperlink{DHH}{DouglasHenriquesHill}).

Since [[gauge theory]] is not just about [[principal bundle]]s, but about principal [[connection on a bundle|bundles with connection]], what matters in physics is not just the topological Spin-, String- and Fivebrane structures, but their refinement to [[schreiber:differential nonabelian cohomology]]. See [[differential fivebrane structure]].

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

\begin{itemize}%
\item [[spin structure]], [[spin connection]];


\item [[string structure]], [[differential string structure]];


\item \textbf{fivebrane structure}, [[differential fivebrane structure]]



\end{itemize}
[[!include higher spin structure - table]]

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

The notion was introduced in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Fivebrane structures]]} , Reviews in mathematical physics, 10 (2009) 1197 (\href{http://arxiv.org/abs/0805.0564}{arXiv:0805.0564})

\end{itemize}
It is briefly mentioned in

\begin{itemize}%
\item [[Chris Douglas]], [[André Henriques]], Michael Hill, \emph{Homological obstructions to string orientations} (\href{http://arxiv.org/abs/0810.2131}{arXiv})

\end{itemize}
Related structures are also mentioned around p. 9 of

\begin{itemize}%
\item [[Edward Witten]], \emph{On Flux Quantization In M-Theory And The Effective Action} (\href{http://arxiv.org/abs/hep-th/9609122}{arXiv:hep-th/9609122})

\end{itemize}
The differential refinement is discussed in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Twisted Differential String and Fivebrane Structures]]} (\href{http://arxiv.org/abs/0910.4001}{arXiv:0910.4001})

\end{itemize}
and

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech Cocycles for Differential characteristic Classes]]} (\href{http://arxiv.org/abs/1011.4735}{arXiv:1011.4735})

\end{itemize}
Articles that use Fivebrane structures include

\begin{itemize}%
\item Boris Botvinnik, [[Mohammed Labbi]], \emph{Highly connected manifolds of positive $p$-curvature}, Transactions of the AMS, Trans. Amer. Math. Soc. 366 (2014), 3405-3424 (\href{http://arxiv.org/abs/1201.1849}{arXiv:1201.1849}, \href{https://doi.org/10.1090/S0002-9947-2014-05939-4}{doi:10.1090/S0002-9947-2014-05939-4})

\end{itemize}
[[!redirects fivebrane structure]]

[[!redirects Fivebrane structures]]



\end{document}