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\begin{document}

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\section*{transgression of bundle gerbes}

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

\noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{transgression_as_a_functor}{Transgression as a functor}\dotfill \pageref*{transgression_as_a_functor} \linebreak
\noindent\hyperlink{application_spin_structures_and_loop_space_orientation}{Application: Spin structures and loop space orientation}\dotfill \pageref*{application_spin_structures_and_loop_space_orientation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

Let $X$ be a [[smooth manifold]], Write $L X = C^\infty(S^1,X)$ for the [[free loop space object|free loop space]].

then [[transgression]] gives a map on [[cohomology]]

\begin{displaymath}
\tau : H^k(X) \to H^{k-1}(L X)
\end{displaymath}
\textbf{Example}

$\mathbb{Z}_2$-coefficients, $k = 2$

\begin{displaymath}
\itexarray{
     H^2(X, \mathbb{Z}_2) &\stackrel{\tau}{\to}& H^1(L X, \mathbb{Z}_2)
     \\
     \xi &\mapsto& \tau(\xi)
  }
\end{displaymath}
where $\xi$ is the second [[Stiefel-Whitney class]] we have that $X$ has [[spin structure]] precisely if $\xi = 0$ is the trivial class. This implies of course that also $\tau(\xi)$ vanishes. Atiyah showed that if the [[fundamental group]] $\pi_1(X) = 1$ of $X$ vanishes, i.e. if $X$ is a [[simply connected space]], that the also the converse holds: $X$ is spin if $\tau(\xi)$ vanishes in the cohomology of the loop space.

\textbf{Questions}

\begin{enumerate}%
\item What is the relation between $\xi$ and $\tau(\xi)$ in general, that would make $\tau$ a bijection.


\item What are relation between trivializations of $\xi$ and those of $\tau(\xi)$ that would make $\tau$ a [[functor]] -- such that this makes transgression an [[equivalence of categories]].



\end{enumerate}
\hypertarget{transgression_as_a_functor}{}\subsection*{{Transgression as a functor}}\label{transgression_as_a_functor}

Let $A$ be an [[abelian group|abelian]] [[Lie group]]. Write $H^2(X,A)$ for the [[abelian sheaf cohomology]].

we want to realize this as the connected components of a [[2-groupoid]] $Grb_A^\nabla(X)$ of [[bundle gerbe]]s with connection on $X$.

Similarly we want to refine $H^1(L X, A)$ to a groupoid $Bun_A^\nabla(L X)$ of [[connection on a bundle|connections]] on smooth $A$-[[principal bundle]]s.

[[Jean-Luc Brylinski]] and MacLaughlin define a [[functor]]

\begin{displaymath}
L : Grb_{A}^\nabla(X) \to Bun_A^\nabla(L X)
   \,.
\end{displaymath}
by

\begin{displaymath}
\mathcal{G} \mapsto L \mathcal{G}|_{\beta}
   :=
   Hom_{Grb_A^\nabla(S^1)}(\beta^* \mathcal{G}, I_0)
\end{displaymath}
for $\beta \in L X$ and where $I_0$ denotes the trivial gerbe on the circle.

We want to understand the image of this transgression map, i.e. to characterize those bundles over $L X$ that can be obtained by transgression of a gerbe on $X$.

\textbf{Definition} Let $P$ be an $A$-[[principal bundle]] over $L X$, then a \textbf{fusion product} on $P$ is a bundle [[isomorphism]] $\lambda$ that is [[fiber]]wise given for a triple of paths

\begin{displaymath}
\gamma_i : x \to y \,,\;\;\;\;\; i \in \{1,2,3\}
\end{displaymath}
\begin{displaymath}
\lambda_{\gamma_1, \gamma_2,\gamma_3}
  :
  P_{\bar\gamma_2 \star \gamma_1}
  \otimes
  P_{\bar \gamma_3 \star \gamma_2}
  \to 
  P_{\bar \gamma_3 \star \gamma_1}
\end{displaymath}
Brylinske-MacLaughlin have a similar fusion product but over figur-8s of paths. This however gives associativity only up to homotopy. Here we are aiming for a product that is strictly associative.

\textbf{Definition} A connection on the fusion bundle $(P,\lambda)$ is called

\begin{enumerate}%
\item \textbf{compactible} if $\lambda$ is connection-preserving;


\item \textbf{symmetrizing}, if

\begin{displaymath}
R_\pi(\lambda(q_1 \otimes q_2))
  =
  \lambda(R_\pi(q_2) \otimes R_\pi(q_1))
  \,,
\end{displaymath}
where $R_\pi$ is a lift of the

\begin{displaymath}
\itexarray{
    P &\stackrel{}{\to}& P
    \\
    \downarrow && \downarrow
    \\
    L X &\stackrel{r_\pi}{\to}& L X
  }
\end{displaymath}
lifts the loop rotation operation by an angle $\pi$ from loop space to the bundle over loop space.

We can take $R$ to be the [[parallel transport]] of the connection on loop space along the canonical path in loop space that connects a loop to its rotated loop.


\item \textbf{superficial} (German: \emph{oberfl\"a{}chlich} -- this is a joke with translations) if it behaves like a surface holonomy in that

\begin{enumerate}%
\item if $\phi \in L L X$, $\tilde \phi : S^1 \times S^1 \to X$ has rank one, then $Hol_P(\phi) = 1$;


\item if $\phi_1, \phi_2  \in L L X$ such that $\tilde \phi_1, \tilde \phi_2$ are rank-2-homotopic (i.e. think homotopic) then $Hol_p(\phi_1) = Hol_p(\phi_2)$.



\end{enumerate}


\end{enumerate}
\textbf{Definition} An $A$-\textbf{fusion bundle} with connection over $L X$ is an $A$-[[nLab:principal bundle]] over $L X$ with fusion product and compatible, symmetrizing and superficial connection.

\textbf{Lemma} Transgression lifts

\begin{displaymath}
\itexarray{
    Grb_A^\nabla(X) &&\stackrel{\tilde K}{\to}&&
    FusBund_A^\nabla(L X)
    \\
    & {}_{\mathllap{L}}\searrow && \swarrow_{\mathrlap{forget}}
    \\
    && Bun_A^\nabla(L X)
  }
\end{displaymath}
\textbf{Theorem} Lifted transgression $\tilde L$ is an equivalence of categories

\hypertarget{application_spin_structures_and_loop_space_orientation}{}\subsection*{{Application: Spin structures and loop space orientation}}\label{application_spin_structures_and_loop_space_orientation}

Assume $\mathcal{G}$ is the $\mathbb{Z}_2$-[[lifting gerbe]] for [[spin structure]] on $X$ whose characteristic class is

\begin{displaymath}
[\mathcal{G}] = \xi \in H^3(X, \mathbb{Z}_2)
\end{displaymath}
the [[Stiefel-Whitney class]] of $X$. So [[spin structure]]s on $X$ are in corresppndence with trivializations of $\mathcal{G}$.

On the other hand we have that [[orientation]]s of $L X$ correspond to [[section]]s of $L \mathcal{G}$. Inside there are the fusion preserving sections, which by the above are equivalent to trivializations of $\mathcal{G}$.

So we find that in general [[spin structure]]s on $X$ are not in bijection to just all orientations of $L X$, but precisely ot the fusion-compatible ones.

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

\begin{itemize}%
\item [[Konrad Waldorf]], \emph{Transgression to Loop Spaces and its Inverse}

\emph{I: Diffeological Bundles and Fusion Maps} (\href{http://arxiv.org/abs/0911.3212}{arXiv:0911.3212})

\emph{II: Gerbes and Fusion Bundles with Connection} (\href{http://arxiv.org/abs/1004.0031}{arXiv:1004.0031})



\end{itemize}


\end{document}