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\section*{Quadratic Functions in Geometry, Topology, and M-Theory}

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

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

[[!include differential cohomology - contents]]

This entry is about the article

\begin{itemize}%
\item [[Michael Hopkins]], [[Isadore Singer]], \emph{Quadratic Functions in Geometry, Topology,and M-Theory} J. Differential Geom. Volume 70, Number 3 (2005), 329-452. (\href{http://arxiv.org/abs/math.AT/0211216}{arXiv:math.AT/0211216}, \href{https://projecteuclid.org/euclid.jdg/1143642908}{euclid:1143642908})

\end{itemize}
which discusses ([[ordinary differential cohomology|ordinary]]) [[differential cohomology]] refinements of [[generalized (Eilenberg-Steenrod) cohomology]] and uses it to study [[quadratic refinements]] (via [[characteristic element of a bilinear form|characteristic cohomology classes]]) of [[intersection pairings]] $(x,y) \mapsto \int_\Sigma x \cup y$ in [[ordinary cohomology]]. [[mathematics|Mathematically]] this refines the construction of [[Theta characteristics]] to [[ordinary differential cohomology]] and to higher [[intermediate Jacobians]]. [[physics|Physically]] it is motivated from and related to [[self-dual higher gauge theory]] (see there for more) appearing in [[string theory]] and the corresponding [[quantum anomaly|quantum anomalies]]. In particular it makes rigorous the construction (\hyperlink{M5-brane#Witten96}{Witten 96}) of the [[partition function]] of the [[self-dual higher gauge field|self-dual]] [[B-field]] in the [[6d (2,0)-superconformal QFT]] on the [[worldvolume]] of the [[M5-brane]] via [[geometric quantization]] of abelian [[7d Chern-Simons theory]].

The article introduces a systematic general definition for the refinement of any [[generalized (Eilenberg-Steenrod) cohomology]] theory to [[differential cohomology]] (the context for higher [[gauge field]]s in physics) in terms of [[differential function complexes]]. In this construction continuous classifying maps from a [[smooth manifold]] into a [[spectrum]] [[Brown representability theorem|representing]] the given [[cohomology]] are equipped with smooth [[differential form]]s that have under the [[de Rham theorem]] the same class in [[real cohomology]] as the pullback along the classifying map of a collection of given real cocycles on the spectrum.

For a somewhat streamlined account see at at \emph{[[differential cohomology hexagon]]} the section \emph{\href{differential%20cohomology%20diagram#HopkinsSingerCoefficients}{Hopkins-Singer coefficients}}.

A review talk is recorded at

\begin{itemize}%
\item [[Michael Hopkins]], \emph{Differential cohomology for general cohomology theories and one physical motivation, talk at Simons Center Workshop on Dierential Cohomology (2011) (\href{http://scgp.stonybrook.edu/archives/851}{video})}

\end{itemize}
This states that the article ``should have been written'' in terms of [[smooth infinity-stacks]]. For references that do so see for instance at \emph{[[differential cohomology hexagon]]} the \href{differential+cohomology+diagram#References}{list of references}.

The connection of this work to the [[physics]] of the [[electromagnetic field]] and of [[higher gauge field]] in [[string theory]] was later further advertized in

\begin{itemize}%
\item [[Daniel Freed]], \emph{[[Dirac charge quantization and generalized differential cohomology]]} (\href{http://arxiv.org/abs/hep-th/0011220}{arXiv:hep-th/0011220}).

\end{itemize}
To this date generalized [[differential cohomology]] theories keep being studied mostly with motivation from [[string theory]], but the work of Hopkins and Singer has put this subject on solid mathematical ground, and an independent mathematical field of differential cohomology is developing since then. See the list of references at \emph{\href{differential+cohomology#References}{differential cohomology -- References}} and at \emph{\href{differential+cohomology+diagram#References}{differential cohomology hexagon -- References}} .

\hypertarget{some_linked_keywords}{}\section*{{Some linked keywords}}\label{some_linked_keywords}

\noindent\hyperlink{differential_cohomology_2}{Differential cohomology}\dotfill \pageref*{differential_cohomology_2} \linebreak
\noindent\hyperlink{algebraic_topology}{Algebraic topology}\dotfill \pageref*{algebraic_topology} \linebreak
\noindent\hyperlink{gauge_fields}{Gauge fields}\dotfill \pageref*{gauge_fields} \linebreak


\hypertarget{differential_cohomology_2}{}\subsection*{{Differential cohomology}}\label{differential_cohomology_2}

\begin{itemize}%
\item [[differential function complex]]


\item [[differential cohomology]]

\begin{itemize}%
\item [[generalized (Eilenberg-Steenrod) cohomology]]

\begin{itemize}%
\item [[ordinary differential cohomology]]


\item [[differential K-theory]]



\end{itemize}

\item [[Chern character]]

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

\end{itemize}


\end{itemize}

\item [[differential orientation]]

\begin{itemize}%
\item [[differential Thom class]]

\end{itemize}

\item [[fiber integration in differential cohomology]]

\begin{itemize}%
\item [[fiber integration in ordinary differential cohomology]]


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



\end{itemize}


\end{itemize}
\hypertarget{algebraic_topology}{}\subsection*{{Algebraic topology}}\label{algebraic_topology}

\begin{itemize}%
\item [[intersection pairing]], [[quadratic refinement]]


\item [[signature genus]], [[Kervaire invariant]]


\item [[Wu class]], [[integral Wu structure]]



\end{itemize}
\hypertarget{gauge_fields}{}\subsection*{{Gauge fields}}\label{gauge_fields}

\begin{itemize}%
\item [[gauge field]]

\begin{itemize}%
\item [[field strength]]


\item [[quantum anomaly]]

\begin{itemize}%
\item [[Green-Schwarz mechanism]]

\end{itemize}


\end{itemize}

\item [[higher dimensional Chern-Simons theory]]


\item [[self-dual higher gauge theory]]



\end{itemize}
category: reference

[[!redirects Quadratic Functions in Geometry, Topology,and M-Theory]]



\end{document}