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\begin{document}

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\section*{twisted differential c-structure}

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

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil 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{Examples}{Examples}\dotfill \pageref*{Examples} \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}

For $c$ any [[characteristic class]], its [[homotopy fibers]] on [[cocycle]] [[∞-groupoid]]s represent $c$-[[twisted cohomology]] (for instance [[twisted bundle]]s, [[twisted spin structures]], etc.).

If $c$ is refined to a characteristic class $\mathbf{c}$ in [[Smooth∞Grpd]] there may exist further refinements $\hat {\mathbf{c}}$ to [[ordinary differential cohomology]]. The twisted cohomology of these \emph{differential characteristic classes} may be called \emph{twisted differential structures} . For instance \emph{[[differential string structure]]s} . See \hyperlink{Examples}{below} for more examples.

These structures have a natural interpretation and play a natural roles as \emph{[[physical fields]]} (see there for a comprehensive discussion).

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

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]], usually $\mathbf{H} =$ [[Smooth∞Grpd]] or [[SynthDiff∞Grpd]] or the like.

Let $K, G$ be [[∞-group]] objects in $\mathbf{H}$ and let

\begin{displaymath}
\mathbf{c} : \mathbf{B}G \to \mathbf{B}K
\end{displaymath}
be a morphism of their [[delooping]] objects / [[moduli stacks]].

\begin{defn}
\label{}\hypertarget{}{}
For $X \in \mathbf{H}$ any object and $P \to X$ an $K$-[[principal ∞-bundle]] over $X$, the [[∞-groupoid]]

\begin{displaymath}
\mathbf{c}Struc_{[P]}(X)
  :=
  \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{P\}
  \,,
\end{displaymath}
hence the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{c}Struc_{[P]}(X)
    &\to&
    *
    \\
    \downarrow^{\mathrlap{P}} && \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}G)
    &\stackrel{\mathbf{H}(X, \mathbf{c})}{\to}&
    \mathbf{H}(X, \mathbf{B}K)
  }
\end{displaymath}
we may call equivalently

\begin{itemize}%
\item the $\infty$-groupoid of $K$-structures on $P$ (with respect to the given $\mathbf{c}$);


\item the $\infty$-groupoid of $[P]$-twisted $\mathbf{c}$-structures.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
As discussed at \emph{[[twisted cohomology]]}, we may think of an object in $\mathbf{c}Struc_{[P]}(X)$ as a [[section]] (up to [[homotopy]]) $\sigma$

\begin{displaymath}
\itexarray{
    && \mathbf{B}G
    \\
    & {}^{\sigma}\nearrow& \downarrow^{\mathbf{c}}
    \\
    X &\stackrel{g}{\to}& \mathbf{B}K
  }
\end{displaymath}
where we think of $\mathbf{c}$ as being the \textbf{universal twisting $\infty$-bundle} and where $g : X \to \mathbf{B}K$ is a morphism presenting $P$.

\end{remark}
The following definition looks at a differential refinement of this situation.

\begin{defn}
\label{}\hypertarget{}{}
For $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ a [[characteristic class|characteristic map]] in $\mathbf{H}$ and $\hat {\mathbf{c}} : \mathbf{B}G_{\mathrm{conn}}
  \to \mathbf{B}^n U(1)_{\mathrm{conn}}$ its differential refinement, sending [[connections on ∞-bundles]] to [[circle n-bundles with connection]] (see [[∞-Chern-Weil homomorphism]], we may think of this also as an [[extended Lagrangian]] for a [[higher gauge theory]]).

We write $\hat {\mathbf{c}}\mathrm{Struc}_{\mathrm{tw}}(X)$ for the corresponding [[twisted cohomology]],

\begin{displaymath}
\itexarray{
    \hat {\mathbf{c}}Struc_{tw}(X) &\stackrel{tw}{\to}& 
    H^{n+1}_{diff}(X)
    \\
    {}^{\mathllap{\chi}}\downarrow && \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}G_{conn})
     &
     \stackrel{\hat \mathbf{c}}{\to}
     &
    \mathbf{H}(X, \mathbf{B}^n U(1)_{conn})
  }
  \,.
\end{displaymath}
\end{defn}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

Twisted differential $\mathbf{c}$-structures appear in various guises in the [[background gauge field]]s of [[string theory]] application.

\begin{itemize}%
\item [[reduction of structure groups]]

\begin{itemize}%
\item [[orthogonal structure]] / [[Riemannian metric]]; see the discussion at \emph{[[vielbein]]} .


\item [[generalized complex geometry]]


\item [[exceptional generalized geometry]]



\end{itemize}

\item [[lift of structure groups]]


\item Higher differential spin structures

\begin{itemize}%
\item [[twisted spin structure]], [[differential spin structure]]


\item [[twisted spin{\tt \symbol{94}}c-structure]]


\item [[twisted differential string structure]]


\item [[supergravity C-field]]


\item [[twisted differential fivebrane structure]]


\item [[twisted Wu structure]]



\end{itemize}

\item [[p1-structure]]

[[2-framing]]


\item [[w4-structure]]

[[w4-orientation of EO(2)-theory]]


\item [[differential T-duality]]



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

\begin{itemize}%
\item [[reduction and lift of structure groups]]


\item [[field (physics)]]



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

The notion was introduced in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Twisted Differential String and Fivebrane Structures]]}

\end{itemize}
and expanded on in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech cocycles for differential characteristic classes]]}

\end{itemize}
An exposition is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:A higher stacky perspective on Chern-Simons theory]]}

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[twisted smooth cohomology in string theory]]}, Lectures at the ESI-institute (June 2012)

\end{itemize}
A general account is in section 5.2 of

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

\end{itemize}
In

\begin{itemize}%
\item [[Daniel Freed]], [[Constantin Teleman]], \emph{Relative quantum field theory} (\href{http://arxiv.org/abs/1212.1692}{arXiv:1212.1692})

\end{itemize}
it is proposed to call such twisted structures ``relative fields''.

[[!redirects c-structure]]

[[!redirects twisted differential c-structure]] [[!redirects twisted differential c-structures]] [[!redirects tiwsted differential c-structure]]

[[!redirects twisted differential structure]] [[!redirects twisted differential structures]]



\end{document}