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\section*{differential K-theory}

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

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

[[!include differential cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{SimonsSullivanModel}{The Simons-Sullivan model}\dotfill \pageref*{SimonsSullivanModel} \linebreak
\noindent\hyperlink{idea_2}{Idea}\dotfill \pageref*{idea_2} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{the_bunkeschick_model}{The Bunke-Schick model}\dotfill \pageref*{the_bunkeschick_model} \linebreak
\noindent\hyperlink{idea_3}{Idea}\dotfill \pageref*{idea_3} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_hopkinssinger_model}{The Hopkins-Singer model}\dotfill \pageref*{the_hopkinssinger_model} \linebreak
\noindent\hyperlink{more_models_in_smooth_spectra}{More models in smooth spectra}\dotfill \pageref*{more_models_in_smooth_spectra} \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
\noindent\hyperlink{History}{History}\dotfill \pageref*{History} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_index_theory}{Relation to index theory}\dotfill \pageref*{relation_to_index_theory} \linebreak
\noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak
\noindent\hyperlink{further}{Further}\dotfill \pageref*{further} \linebreak


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

\emph{Differential K-theory} is the refinement of the [[generalized (Eilenberg-Steenrod) cohomology]] theory [[K-theory]] to [[differential cohomology]].

In as far as we can think of cocycles in [[K-theory]] as represented by [[vector bundle]]s or [[vectorial bundle]]s, cocycles in differential K-theory may be represented by [[vector bundle]]s [[connection on a bundle|with connection]].

There are various different models that differ in the concrete realization of these cocycles and in their extra properties.

\hypertarget{SimonsSullivanModel}{}\subsection*{{The Simons-Sullivan model}}\label{SimonsSullivanModel}

This section discusses the model presented in (\hyperlink{SimonsSullivan}{SimonsSullivan}).

More details will eventually be at

\begin{itemize}%
\item [[Simons-Sullivan structured bundle]].

\end{itemize}
\hypertarget{idea_2}{}\subsubsection*{{Idea}}\label{idea_2}

In the Simons-Sullivan model cocycles in differential K-theory are represented by ordinary [[vector bundle]]s [[connection on a bundle|with connection]]. The crucial ingredient is that two connections on a vector bundle are taken to be the same representative of a differential K-cocycle if they are related by a [[concordance]] such that the corresponding [[Chern-Simons form]] is exact.

\hypertarget{details}{}\subsubsection*{{Details}}\label{details}

Let $V \to X$ be a complex [[vector bundle]] with [[connection on a bundle|connection]] $\nabla$ and [[curvature]] 2-form

\begin{displaymath}
F = F_\nabla \in \Omega^2(X,End(V))
  \,.
\end{displaymath}
\textbf{Definition}

The \textbf{[[Chern character]]} of $\nabla$ is the inhomogenous [[curvature characteristic form]]

\begin{displaymath}
ch(\nabla) := \sum_{j \in \mathbb{N}}
   k_j
   tr( F_\nabla \wedge \cdots \wedge F_\nabla)
  \;\;
  \in \Omega^{2 \bullet}(X)
  \,,
\end{displaymath}
where on the right we have $j$ wedge factors of the [[curvature]] .

\textbf{Definition}

Let $(V,\nabla)$ and $(V',\nabla')$ be two complex vector bundles with connection.

A [[Chern-Simons form]] for this pair is a differential form

\begin{displaymath}
CS(\nabla,\nabla') + d \omega \in
  \Omega^{2 \bullet + 1}(X)
\end{displaymath}
obtained from the [[concordance]] bundle $\bar V \to X \times [0,1]$ given by [[pullback]] along $X \times [0,1] \to X$ equipped with a connection $\bar \nabla$ such that \ldots{}, by

\begin{displaymath}
CS(\nabla,\nabla') =
  \int_0^1 \psi_t^* (\iota_{\partial/\partial t} ch(\bar \nabla))
  + d (...)
  \,.
\end{displaymath}
\textbf{Proposition} This is indeed well defined in that it is independent of the chosen [[concordance]], up to an exact term.

\textbf{Definition}

A \textbf{structured bundle} in the sense of the Simons-Sullivan model is a complex vector bundle $V$ equipped with the equivalence class $[\nabla]$ of a connection under the equivalence relation that identifies two connections $\nabla$ and $\nabla'$ if their Chern-Simons form $CS(\nabla,\nabla')$ is exact.

Two structured bundles are isomorphic if there is a vector bundle isomorphism under which the two equivalence classes of connections are identified.

\textbf{Definition}

Let $Struc(X)$ be the set of [[isomorphism]] classes of structured bundles on $X$.

Under [[direct sum]] and [[tensor product]] of vector bundles, this becomes a commutatve [[rig]].

Let

\begin{displaymath}
\hat K(X) := K(Struct(X))
\end{displaymath}
be the additive [[group completion]] of this rig as usual in [[K-theory]].

So as an additive group $\hat K(X)$ is the quotient of the [[monoid]] induced by [[direct sum]] on pairs $(V,W)$ of isomorphism classes in $Struc(X)$, modulo the sub-monoid consisting of pairs $(V,V)$.

Hence the pair $(V,0)$ is the additive inverse to $(0,V)$ and $(V,W)$ may be written as $V - W$.

\textbf{Theorem}

$\hat K(X)$ is indeed a [[differential cohomology]] refinement of ordinary K-theory $K(X)$ of $X$ (i.e. of the 0th [[cohomology group]] of [[K-theory spectrum|K-cohomology]]).

Moreover\ldots{}

\hypertarget{the_bunkeschick_model}{}\subsection*{{The Bunke-Schick model}}\label{the_bunkeschick_model}

\hypertarget{idea_3}{}\subsubsection*{{Idea}}\label{idea_3}

[[Uli Bunke]] and [[Thomas Schick]] developed in a series of articles a differential-geometric cocycle model of differential K-theory where cocycles are given by smooth families of [[Dirac operator]]s.

See the reference \href{BunkeSchickReferences}{below}.

\hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties}

The restriction of the cocycles in the Bunke-Schick model to those whose ``auxialiary form'' $\omega$ vanishes reproduces the Simons-Sullivan model above.

\hypertarget{the_hopkinssinger_model}{}\subsection*{{The Hopkins-Singer model}}\label{the_hopkinssinger_model}

See at

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


\item \emph{\href{differential+cohomology+diagram#HSDifferentialKU}{differential cohomology diagram -- Hopkins-Singer differential KU}}



\end{itemize}
\hypertarget{more_models_in_smooth_spectra}{}\subsection*{{More models in smooth spectra}}\label{more_models_in_smooth_spectra}

See at \emph{\href{differential+cohomology+diagram#DifferentialKTheory}{Differential cohomology diagram -- Differential K-theory}}.

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

\begin{itemize}%
\item In [[gauge theory]] gauge fields are modeled by cocycles in [[differential cohomology]]. The field modeled by differential K-theory is the [[RR-field]]. A kind of [[integral transform]] acting on differential K-theory classes is [[T-duality]].

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

\begin{itemize}%
\item [[K-theory]]


\item [[topological K-theory]]


\item [[twisted K-theory]]


\item \textbf{differential K-theory}

\begin{itemize}%
\item [[fiber integration in differential K-theory]]


\item [[algebraic K-theory of smooth manifolds]]



\end{itemize}

\item [[twisted differential K-theory]]


\item [[differential algebraic K-theory]]



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

\hypertarget{History}{}\subsubsection*{{History}}\label{History}

An early sketch of a definition, motivated by the description of [[D-brane charge]] in [[string theory]], is in

\begin{itemize}%
\item [[Daniel Freed]], \emph{[[Dirac charge quantization and generalized differential cohomology]]}, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129--194 (\href{http://arxiv.org/abs/hep-th/0011220}{arXiv:hep-th/0011220})


\item [[Daniel Freed]], [[Michael Hopkins]], \emph{On Ramond-Ramond fields and K-theory}, JHEP (2000) 44, 14 (\href{http://arxiv.org/abs/hep-th/0002027}{arXiv:hep-th/0002027})



\end{itemize}
Then the general construction of [[differential cohomology]] theories via [[differential function complexes]] of

\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})

\end{itemize}
(motivated in turn by [[7d Chern-Simons theory]] and the [[M5-brane]] [[partition function]])

provides in particular a model for differential K-theory.

For more historical remarks see section 1.6 of

\begin{itemize}%
\item [[Daniel Freed]], [[John Lott]], \emph{An index theorem in differential K-theory}, Geometry and Topology 14 (2010) (\href{http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf}{pdf})

\end{itemize}
A discussion of more models and their relation in the context of [[cohesive homotopy type theory]] and the [[differential cohomology hexagon]] then appears in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], section 6 of \emph{Differential cohomology theories as sheaves of spectra} (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188})

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

A review is in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Schick]], \emph{Differential K-theory. A survey} (\href{http://arxiv.org/abs/1011.6663}{arXiv:1011.6663}).

\end{itemize}
The Simons-Sullivan model is due to

\begin{itemize}%
\item [[James Simons]], [[Dennis Sullivan]], \emph{Structured vector bundles define differential K-theory} (\href{http://arxiv.org/abs/0810.4935}{arXiv:0810.4935})

\end{itemize}
The basic article for the Bunke-Schick model is

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Schick]], \emph{Smooth K-Theory} (\href{http://arxiv.org/abs/0707.0046}{arXiv:0707.0046})

\end{itemize}
A survey talk is

\begin{itemize}%
\item [[Ulrich Bunke]], \emph{Differential cohomology in geometry and analysis} (\href{http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/Vortrag-Erlangen.pdf}{pdf slides})

\end{itemize}
Differential [[KO-theory]] is studied in

\begin{itemize}%
\item Daniel Grady, [[Hisham Sati]], \emph{Differential KO-theory: Constructions, computations, and applications} (\href{https://arxiv.org/abs/1809.07059}{arXiv:1809.07059})

\end{itemize}
Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] in

\begin{itemize}%
\item Daniel Grady, [[Hisham Sati]], \emph{Twisted differential KO-theory} (\href{https://arxiv.org/abs/1905.09085}{arXiv:1905.09085})

\end{itemize}
The differential version of [[equivariant K-theory]] is in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Schick]], \emph{Differential orbifold $K$-Theory} (\href{http://arxiv.org/abs/0905.4181}{arXiv:0905.4181})

\end{itemize}
The equivalence of these models with the respective special case of the general construction in

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

\end{itemize}
in terms of [[differential function complexes]] is in

\begin{itemize}%
\item Kevin Klonoff, \emph{An Index Theorem in Differential K-Theory} PdD thesis (2008) (\href{http://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf}{pdf})

\end{itemize}
(assuming the existence of a universal connection, which is not strictly proven) and

\begin{itemize}%
\item Michael L. Ortiz, \emph{Differential Equivariant K-Theory} (\href{http://arxiv.org/abs/0905.0476}{arXiv:0905.0476})

\end{itemize}
(not needing that assumption).

A construction of [[differential cobordism cohomology]] theory in terms of explicit geometric cocycles is in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup \emph{Landweber exact formal group laws and smooth cohomology theories} (\href{http://arxiv.org/abs/0711.1134}{arXiv:0711.1134})

\end{itemize}
By tensoring this with the suitable ring, this also gives a model for differential K-theory, as well as for [[differential elliptic cohomology]].

A variant of this definition with the advantage that there is a natural morphism to [[Cheeger-Simons differential characters]] refining the total [[Chern class]] is (as opposed to the [[Chern character]]) is presented in

\begin{itemize}%
\item Alain Berthomieu, \emph{A version of smooth K-theory adapted to the total Chern class} (\href{http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.4728v1.pdf}{pdf})

\end{itemize}
Discussion for the [[odd Chern character]] is in

\begin{itemize}%
\item [[Thomas Tradler]], [[Scott Wilson]], [[Mahmoud Zeinalian]], \emph{An Elementary Differential Extension of Odd K-theory}, J. of K-theory, K-theory and its Applications to Algebra, Geometry, Analysis and Topology, (\href{http://arxiv.org/abs/1211.4477}{arXiv:1211.4477})


\item [[Scott Wilson]], \emph{A loop group extension of the odd Chern character} (\href{http://arxiv.org/abs/1311.6393}{arXiv:1311.6393})



\end{itemize}
\hypertarget{relation_to_index_theory}{}\subsubsection*{{Relation to index theory}}\label{relation_to_index_theory}

Relation to [[index theory]]:

\begin{itemize}%
\item Kevin Klonoff, \emph{An Index Theorem in Differential K-Theory} PdD thesis (2008) (\href{http://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf}{pdf})


\item [[Daniel Freed]], [[John Lott]], \emph{An index theorem in differential K-theory}, Geometry and Topology 14 (2010) (\href{http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf}{pdf})



\end{itemize}
See also the references at \emph{[[fiber integration in differential K-theory]]}.

\hypertarget{in_string_theory}{}\subsubsection*{{In string theory}}\label{in_string_theory}

A survey of the role of differential $K$-theory in [[quantum field theory]] and [[string theory]] is in

\begin{itemize}%
\item [[Daniel Freed]], \emph{K-Theory in Quantum Field Theory}, Current developments in Mathematics (2001) International Press Somerville (\href{http://arxiv.org/abs/math-ph/0206031}{arXiv:math-ph/0206031})

\end{itemize}
The operation of [[T-duality]] on hypothetical [[twisted differential K-theory]] is discussed in

\begin{itemize}%
\item [[Alexander Kahle]], [[Alessandro Valentino]], \emph{[[T-Duality and Differential K-Theory]]}

\end{itemize}
Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type II string theory]] (see also \href{D-brane#ReferencesKTheoryDescription}{there}):

\begin{itemize}%
\item Daniel Grady, [[Hisham Sati]], \emph{Ramond-Ramond fields and twisted differential K-theory} (\href{https://arxiv.org/abs/1903.08843}{arXiv:1903.08843})

\end{itemize}
Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type I string theory]] (on [[orientifolds]]):

\begin{itemize}%
\item Daniel Grady, [[Hisham Sati]], \emph{Twisted differential KO-theory} (\href{https://arxiv.org/abs/1905.09085}{arXiv:1905.09085})

\end{itemize}
\hypertarget{further}{}\subsubsection*{{Further}}\label{further}

See also

\begin{itemize}%
\item Thomas Tradler, Scott O. Wilson, [[Mahmoud Zeinalian]], \emph{Loop Differential K-theory} (\href{http://arxiv.org/abs/1201.4593}{arXiv:1201.4593})

\end{itemize}


\end{document}