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\section*{fiber integration in ordinary differential cohomology}

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

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

[[!include differential cohomology - contents]]

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration 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{differential_orientation}{Differential orientation}\dotfill \pageref*{differential_orientation} \linebreak
\noindent\hyperlink{FiberIntegration}{Via differential Thom cocycles}\dotfill \pageref*{FiberIntegration} \linebreak
\noindent\hyperlink{ViaDeligneComplex}{In terms of Deligne cocycles}\dotfill \pageref*{ViaDeligneComplex} \linebreak
\noindent\hyperlink{ViaSmoothHomotopyType}{In terms of smooth homotopy types}\dotfill \pageref*{ViaSmoothHomotopyType} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{abstract_formulation}{Abstract formulation}\dotfill \pageref*{abstract_formulation} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{CSInHigherCodimension}{$\infty$-Chern-Simons functionals in higher codimension}\dotfill \pageref*{CSInHigherCodimension} \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 special case of [[fiber integration in differential cohomology]] for \emph{[[ordinary differential cohomology]]} is the partial [[higher parallel transport|higher]] [[holonomy]] operation for [[circle n-bundles with connection]]:

for $Y \to X$ a [[bundle]] of [[compact space|compact]] [[smooth manifolds]] $S$ of [[dimension]] $k$ and $[\nabla] \in H_{diff}^n(Y)$ a class in [[ordinary differential cohomology]] of degree $n$ on $Y$, its [[fiber integration]]

\begin{displaymath}
\left[\exp(i \int_{Y/X} \nabla)\right] \in H^{n-k}_{diff}(X)
\end{displaymath}
is a differential cohomology class on $X$ of degree $k$ less.

In the particular case that $X = *$ is the [[point]] and $dim Y = k = n-1$ the element

\begin{displaymath}
\exp(i \int_{Y} \nabla) \in H^{1}_{diff}(*) \simeq U(1)
\end{displaymath}
is the [[higher parallel transport|higher]] [[holonomy]] of $\nabla$ over $Y$.

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

\hypertarget{differential_orientation}{}\subsubsection*{{Differential orientation}}\label{differential_orientation}

The operation of [[fiber integration]] in [[generalized (Eilenberg-Steenrod) cohomology]] requires a choice of [[orientation in generalized cohomology]]. For fiber integration in [[differential cohomology]] this is to be refined to a \emph{differential orientation} .

Accordingly, instead of a [[Thom class]] there is a \emph{differential Thom class} .

\begin{defn}
\label{DifferentialThomCocycle}\hypertarget{DifferentialThomCocycle}{}
For $X$ a [[compact space|compact]] [[smooth manifold]] and $V \to X$ a smooth real [[vector bundle]] of [[rank]] $k$ a \textbf{differential Thom cocycle} on $V$ is

\begin{itemize}%
\item a [[compact support|compactly supported]] [[cocycle]] $\hat \omega$ in the [[ordinary differential cohomology]] of degree $k$ of $V$;


\item such that for each $x \in X$ we have

\begin{displaymath}
\int_{V_x} \omega = \pm 1
  \,.
\end{displaymath}


\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The underlying class $[\hat \omega] \in H^{k}_{compact}(V, \mathbb{Z})$ in [[compact support|compactly supported]] [[integral cohomology]] is an ordinary [[Thom class]] for $V$.

\end{remark}
\begin{defn}
\label{DifferentialOrientation}\hypertarget{DifferentialOrientation}{}
Let $p : X \to Y$ be a [[smooth function]] of [[smooth manifold]]s.

An \textbf{$H \mathbb{Z}_{diff}$-orientation} on $p$ is

\begin{enumerate}%
\item A factorization through an [[embedding of smooth manifolds]]

\begin{displaymath}
p : X \hookrightarrow Y \times \mathbb{R}^N \stackrel{}{\to} Y
\end{displaymath}
for some $N \in \mathbb{N}$;


\item a [[tubular neighbourhood]] $W \hookrightarrow Y \times \mathbb{R}^N$ of $X$;


\item a differential Thom cocycle, def. \ref{DifferentialThomCocycle}, $U$ on $W \to X$.



\end{enumerate}
\end{defn}
This appears as (\hyperlink{HopkinsSinger}{HopkinsSinger, def. 2.9}).

\hypertarget{FiberIntegration}{}\subsubsection*{{Via differential Thom cocycles}}\label{FiberIntegration}

Write $H^n_{diff}(-)$ for [[ordinary differential cohomology]]. For any choice of presentation, there is a fairly evident fiber integration of [[compact support|compactly supported]] [[cocycles]] along trivial [[Cartesian space]] bundles $Y \times \mathbb{R}^N \to Y$ over a [[compact space|compact]] $Y$:

\begin{displaymath}
\int_{\mathbb{R}^N} 
   : H^{n+N}_{diff,cpt}(Y \times \mathbb{R}^n)
  \to
  H^n_{diff}(Y)
  \,.
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
Let $X \to Y$ be a [[smooth function]] equipped with differential $H\mathbb{Z}$-orientation $U$, def. \ref{DifferentialOrientation}. Then the corresponding \textbf{fiber integration} of ordinary differential cohomology is the composite

\begin{displaymath}
\int_{X/Y} :
   H_{diff}^{n+k}(X)
    \stackrel{(-)\cup U}{\to}
   H_{diff, cpt}^{n+N}(X \times \mathbb{R}^N)
    \stackrel{\int_{\mathbb{R}^N}}{\to}
   H_{diff}^n(Y)
  \,.
\end{displaymath}
\end{defn}
This appears as (\hyperlink{HopkinsSinger}{HopkinsSinger, def. 3.11}).

\hypertarget{ViaDeligneComplex}{}\subsubsection*{{In terms of Deligne cocycles}}\label{ViaDeligneComplex}

We discuss an explicit formula for fiber integration along [[product]]-bundles with [[compact topological space|compact]] [[fibers]] in terms of [[Deligne complex]], following (\hyperlink{GomiTerashima00}{Gomi-Terashima 00}).

For $X$ a [[smooth manifold]], write $\mathbf{H}(X, \mathbf{B}^n U(1)_{conn})$ for the [[Deligne complex]] in degree $(n+1)$ over $X$.

\begin{defn}
\label{GomiFiberIntegration}\hypertarget{GomiFiberIntegration}{}
Let $X$ be a [[paracompact topological space|paracompact]] [[smooth manifold]] and let $F$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$ without [[boundary]]. Then there is a morphism

\begin{displaymath}
\int_F 
    \;\colon\;
  \mathbf{H}(X, \mathbf{B}^n U(1)_{conn})
    \to 
  \mathbf{H}(X, \mathbf{B}^{n-k} U(1)_{conn})
\end{displaymath}
given by (\ldots{})

\end{defn}
(\hyperlink{GomiTerashima00}{Gomi-Terashima 00, section 2, corollary 3.2})

\hypertarget{ViaSmoothHomotopyType}{}\subsubsection*{{In terms of smooth homotopy types}}\label{ViaSmoothHomotopyType}

The \hyperlink{ViaDeligneComplex}{above} formulation of fiber integration in ordinary differential cohomology serves as a presentation for a more abstract construction in [[smooth infinity-groupoid|smooth homotopy theory]].

Let $\mathbf{H} \coloneqq$ [[Smooth∞Grpd]] be the ambient [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]]/smooth [[∞-stacks]]. As discussed there, the [[Deligne complex]], being a [[sheaf]] of [[chain complexes]] of [[abelian groups]], presents under the [[Dold-Kan correspondence]] a [[simplicial presheaf]] on the [[site]] [[CartSp]], which in turn presents an object

\begin{displaymath}
\mathbf{B}^n U(1)_{conn} \in \mathbf{H}
  \,,
\end{displaymath}
discussed \emph{\href{http://nlab.mathforge.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#DifferentialCohomology}{here}}: the smooth [[moduli ∞-stack]] of [[circle n-bundles with connection]].

Let now $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k \in \mathbb{N}$ without [[boundary]]. There is the \href{infinity,1%29-topos#ClosedMonoidalStructure}{internal hom in an (infinity,1)-topos}

\begin{displaymath}
[\Sigma_k, \mathbf{B}^n U(1)_{conn}] \in \mathbf{H}
  \,,
\end{displaymath}
which is the smooth moduli $n$-stack of circle $n$-connections \emph{on} $\Sigma_k$.

\begin{prop}
\label{}\hypertarget{}{}
For all $k \leq n$ there is a natural morphism

\begin{displaymath}
\exp(2\pi i\int_\Sigma(-)) 
  \; \colon \;
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
  \to 
  \mathbf{B}^{n-k} U(1)_{conn}
  \;\;\;
  \in 
  \mathbf{H}
  \,.
\end{displaymath}
which for $U \in$ [[SmthMfd]] a smooth test manifold sends $n$-connections on $\Sigma_k$ on $U \times \Sigma_k$ to the $(n-k)$-connection on $U$ which is their fiber integration over $\Sigma_k$.

\end{prop}
(\hyperlink{FiorenzaSatiSchreiber12}{Fiorenza-Sati-Schreiber 12})

\begin{proof}
To see this, observe that

\begin{enumerate}%
\item by definition $\mathbf{H}(U, [\Sigma_k, \mathbf{B}^n U(1)_{conn}]) \simeq \mathbf{H}(U \times \Sigma_k, \mathbf{B}^n U(1)_{conn})$;


\item if $\{U_i \to \Sigma_k\}$ is a fixed [[good open cover]] of $\Sigma_k$, then $\{U \times U_i \to U \times \Sigma_k\}$ is also a good open cover, for every $U \in$ [[CartSp]];


\item hence the [[Cech nerve]] $C(\{U \times U_i\})$ is a natural (functorial in $U \in CartSp$) [[cofibrant object]] [[resolution]] of $U \times \Sigma_k$ in the projective local [[model structure on simplicial presheaves]] $[CartSp^{op}, sSet]_{proj,loc}$ which [[presentable (infinity,1)-category|presents]] $\mathbf{H} =$[[Smooth∞Grpd]] (as discussed there);


\item the (image under the [[Dold-Kan correspondence]]) of the [[Deligne complex]] $\mathbb{Z}(n+1)^\infty_D$ is a is fibrant in this model structure (since every circle $n$-bundle is trivializable over a [[contractible topological space|contractible space]] $U \in$ [[CartSp]]).



\end{enumerate}
This means that a presentation of $[\Sigma_k, \mathbf{B}^n U(1)_{conn}]$ by an object of $[CartSp^{op}, sSet]_{proj,loc}$ is given by the [[simplicial presheaf]]

\begin{displaymath}
U \mapsto DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\}))
\end{displaymath}
that sends $U$ to the [[Cech cohomology|Cech]]-[[Deligne complex|Deligne]] [[hypercohomology]] [[chain complex]] with respect to the cover $\{U \times U_i \to U \times \Sigma_k\}$.

On this def. \ref{GomiFiberIntegration} provides a morphism of simplicial sets

\begin{displaymath}
DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\}))
  \to
  DK \mathbb{Z}(n+1)^\infty_D(U)
\end{displaymath}
which one directly sees is natural in $U$, hence extends to a morphism of simplicial presheaves, which in turn presents the desired morphism in $\mathbf{H}$.

\end{proof}
Applications are to

\begin{itemize}%
\item [[transgression]]


\item [[double dimensional reduction]]



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

(\ldots{})

\hypertarget{abstract_formulation}{}\subsubsection*{{Abstract formulation}}\label{abstract_formulation}

At least the fiber integration all the way to the point exists on general grounds for the  in any [[cohesive (∞,1)-topos]]: the general abstract formulation is in the section \emph{} and the implementation in [[smooth ∞-groupoid]]s is in the section \emph{} .

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{CSInHigherCodimension}{}\subsubsection*{{$\infty$-Chern-Simons functionals in higher codimension}}\label{CSInHigherCodimension}

(\ldots{})

Differential universal characteristic class / extended $\infty$-Chern-Simons Lagrangian:

\begin{displaymath}
\hat \mathbf{c} : \mathbf{B}G_{conn}  \to \mathbf{B}^{n}U(1)_{conn}
\end{displaymath}
moduli $\infty$-stack of [[higher gauge fields]] on a given $\Sigma_k$:

\begin{displaymath}
[\Sigma_k, \mathbf{B}G_{conn}] \in \mathbf{H}
\end{displaymath}
Lagrangian of $\hat \mathbf{c}$-Chern-Simons theory:

\begin{displaymath}
[\Sigma_k, \hat \mathbf{c}] 
   : 
  [\Sigma_k, \mathbf{B}G_{conn}]
   \to
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
\end{displaymath}
extended action functional of $\hat \mathbf{c}$-Chern-Simons theory in codimension $(n-k)$

\begin{displaymath}
\exp(2 \pi i \int_{\Sigma_k}
  [\Sigma_k, \hat \mathbf{c}] 
  )
   : 
  [\Sigma_k, \mathbf{B}G_{conn}]
    \stackrel{[\Sigma_k, \hat \mathbf{c}]}{\to}
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}] 
   \stackrel{\exp(2 \pi i\int_{\Sigma_k} (-))}{\to}
  \mathbf{B}^{n-k} U(1)_{conn}
  \,.
\end{displaymath}
(\ldots{})

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

\begin{itemize}%
\item [[higher parallel transport]], [[higher holonomy]]


\item [[fiber integration in differential K-theory]]



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

A discussion in the general sense of [[fiber integration]] in [[generalized (Eilenberg-Steenrod) cohomology]] is in section 3.4 of

\begin{itemize}%
\item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology,and M-Theory]]}

\end{itemize}
and around prop. 2.1 (in the context of [[Chern-Simons theory]]) in

\begin{itemize}%
\item [[Daniel Freed]], \emph{Classical Chern-Simons theory} II. Special issue for S. S. Chern. Houston J. Math. 28 (2002), no. 2, 293--310.

\end{itemize}
Explicit formulas for fiber integration of \emph{cocycles} in [[Cech cohomology|Cech]]-[[Deligne cohomology]] are given in

\begin{itemize}%
\item [[Kiyonori Gomi]] and Yuji Terashima, \emph{A Fiber Integration Formula for the Smooth Deligne Cohomology} International Mathematics Research Notices 2000, No. 13 (\href{http://imrn.oxfordjournals.org/content/2000/13/699.full.pdf}{pdf}, \href{http://numr.wdfiles.com/local--files/differential-cohomology/gomi-terashima.pdf}{pdf})

\end{itemize}
and their generalization from [[higher holonomy]] to [[higher parallel transport]] in

\begin{itemize}%
\item [[Kiyonori Gomi]] and Yuji Terashima, \emph{Higher dimensional parallel transport} Mathematical Research Letters 8, 25--33 (2001) (\href{http://intlpress.com/_newsite/site/pub/files/_fulltext/journals/mrl/2001/0008/0001/MRL-2001-0008-0001-A-004.pdf}{pdf})

\end{itemize}
and

\begin{itemize}%
\item David Lipsky, \emph{Cocycle constructions for topological field theories} (2010) ([[LipskyThesis.pdf:file]])

\end{itemize}
See also

\begin{itemize}%
\item [[Johan Dupont]], Rune Ljungmann, \emph{Integration of simplicial forms and Deligne cohomology} Math. Scand. 97 (2005), 11--39 (\href{http://www.mscand.dk/article.php?id=897}{pdf})

\end{itemize}
The observation that the construction in \hyperlink{GomiTerashima00}{Gomi-Terashima 00} induces refines to [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack|higher moduli stacks]] is discussed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Extended higher cup-product Chern-Simons theories]]}, Journal of Geometry and Physics Volume 74, 2013, Pages 130--163 (\href{http://arxiv.org/abs/1207.5449}{arXiv:1207.5449})

\end{itemize}
for the case without boundary and for the general case in

\begin{itemize}%
\item [[Urs Schreiber]] et al. \emph{[[schreiber:Local prequantum field theory]]}.

\end{itemize}
[[!redirects differential Thom class]]

[[!redirects integration in ordinary differential cohomology]]



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