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\section*{Chern character}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{KTheory}{For vector bundles and topological K-theory}\dotfill \pageref*{KTheory} \linebreak
\noindent\hyperlink{for_spectra_and_generalized_cohomology_theories}{For spectra and generalized cohomology theories}\dotfill \pageref*{for_spectra_and_generalized_cohomology_theories} \linebreak
\noindent\hyperlink{for_cohesive_stable_homotopy_types}{For cohesive stable homotopy types}\dotfill \pageref*{for_cohesive_stable_homotopy_types} \linebreak
\noindent\hyperlink{in_terms_of_cyclic_homology}{In terms of cyclic homology}\dotfill \pageref*{in_terms_of_cyclic_homology} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{pushforward_and_grothendieckriemannroch_theorem}{Push-forward and Grothendieck-Riemann-Roch theorem}\dotfill \pageref*{pushforward_and_grothendieckriemannroch_theorem} \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}

Traditionally, in the strict sense of the term, the \emph{Chern character} is a [[universal characteristic class]] of [[vector bundles]] or equivalently of their [[topological K-theory]] classes, which is a [[rational number|rational]] combination of all [[Chern classes]].

This is a special case of the following more general construction ([[Quadratic Functions in Geometry, Topology,and M-Theory|Hopkins-Singer 02, section 4.8]]):

for $E$ a [[spectrum]] [[Brown representability theorem|representing]] a [[generalized (Eilenberg-Steenrod) cohomology theory]] there is a canonical [[localization of spectra|localization]] map

\begin{displaymath}
ch_E \;\colon\; E \longrightarrow E \wedge H\mathbb{R}
\end{displaymath}
to the [[smash product]] with the [[Eilenberg-MacLane spectrum]] over the [[real numbers]]. This represents the $E$-Chern character (see also \hyperlink{BunkeGepner13}{Bunke-Gepner 13, around def. 2.1}).

In the case that $E =$ [[KU]] this reproduces the traditional Chern character. (In which case this is a map from a [[complex oriented cohomology theory]] of [[chromatic level]] 1 to chromatic level 0. More generally one can also consider [[higher chromatic Chern characters]] that take values not in [[ordinary cohomology]] but in some cohomology theory of higher [[chromatic level]]. See at \emph{[[higher chromatic Chern character]]} for more on this.)

The Chern character $ch_E$ may be used to define [[differential cohomology]] refinements $\hat E$ of the [[cohomology theory]] $E$ by choosing a [[differential form]]-model for $E \wedge H\mathbb{R}$ (\hyperlink{HopkinsSinger02}{Hopkins-Singer 02}, see also at \emph{[[differential function complex]]}). In that case $ch_E$ is the [[real cohomology]] class associated to a \emph{chern character differential form} $CH_E$ via the [[de Rham theorem]]. Here $CH_E$ has the interpretation of being the [[curvature forms]] of the [[differential cohomology]] [[cocycles]] thought of as [[connection on a principal ∞-bundle|∞-connections]].

This may be turned around (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-V\"o{}lkl 13, prop. .3.5}): given any refinement $\hat E$ of $E$ in a [[tangent cohesive (∞,1)-topos]] $T \mathbf{H}$, then it is induced from homotopy pullback of its \href{cohesive+%28infinity%2C1%29-topos+--+structures#deRhamCohomology}{de Rham coefficients} along a Chern character map

\begin{displaymath}
ch_E = \Pi \theta_{\hat E} \;\colon\; E \simeq \Pi(\hat E) \longrightarrow \Pi \flat_{dR} \hat E
  \,,
\end{displaymath}
where $\Pi$ is the [[shape modality]] and $\theta_E$ the [[Maurer-Cartan form]] of $E$. This reproduces the above definition for ordinary differential form models, see at \emph{\hyperlink{differential+cohomology+diagram#HopkinsSingerCoefficients}{differential cohomology diagram -- Hopkins-Singer coefficients}}.

But more generally, given for instance a [[K(n)-local spectrum|K(n)-localization]] $E \longrightarrow L_{K(n)} E$ then any choice of [[cohesion|cohesive]] refinement of $L_{K(n)} E$ (i.e. lift through the [[unit of a monad|unit]] of the [[shape modality]] $\Pi$) which is in the kernel of $\flat$ yields a generalized differential cohomology theory $\hat E$ whose intrinsic Chern-character $\Pi \theta_{\hat E}$ is the $K(n)$-localization. See at \emph{\href{differential+cohomology+diagram#ChernCharacterAndFractureSquares}{differential cohomology diagram -- Chern character and differential fracture}}.

In words this is summarized succintly as: \emph{The Chern character is the [[shape modality|shape]] of the [[Maurer-Cartan form]].}

In the context of [[algebraic K-theory]] Chern characters appear at \emph{[[Beilinson regulators]]}. There are analogues in algebraic geometry (e.g. a Chern character between [[Chow groups]] and [[algebraic K-theory]]) and in [[noncommutative geometry]] (Chern-Connes character) where the role of usual cohomology is taken by some variant of cyclic cohomology.

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

\hypertarget{KTheory}{}\subsubsection*{{For vector bundles and topological K-theory}}\label{KTheory}

The classical theory of the Chern character applies to the [[spectrum]] of complex [[K-theory]], $E = KU$. In this case, the Chern character is made up from Chern classes: each characteristic class is by Chern-Weil theory in the image of a certain element in the Weil algebra via taking the class of evaluation at the [[curvature]] operator for some choice of a connection. Consider the symmetric functions in $n$ variables $t_1,\ldots, t_n$ and let the Chern classes of a complex vector bundle $\xi$ (representing a complex K-theory class) be $c_1,\ldots, c_n$. Define the formal power series

\begin{displaymath}
\phi = \phi^n(t_1,\ldots, t_n) = e^{t_1}+\ldots+e^{t_n}= \sum_{k=0}^\infty \frac{1}{k!} (t_1^k+\ldots+t_n^k)
\end{displaymath}
Then $ch(\chi) = \phi(c_1,\ldots,c_n)$.

Let us describe this a bit differently. The cocycle $H^0(X,KU)$ may be represented by a complex [[vector bundle]], and the image of this cocycle under the Chern-character is the class in even-graded real cohomology that is represented (under the [[deRham theorem]] isomorphism of deRham cohomology with real cohomology) by the even graded closed [[differential 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

\begin{itemize}%
\item $\nabla$ is any chosen [[connection on a bundle|connection]] on the vector bundle;


\item $F = F_\nabla \in \Omega^2(X,End(V))$ is the [[curvature]] of this connection;


\item $k_j \in \mathbb{R}$ are normalization constants, $k_j = \frac{1}{j!} \left( \frac{1}{2\pi i}\right)^j$;


\item the trace of the wedge products produces the [[curvature characteristic form]]s.



\end{itemize}
The Chern character applied to the [[Whitney sum]] of two vector bundles is a sum of the Chern characters for the two: $ch(\xi\oplus \eta) = ch(\chi)+ch(\eta)$ and it is multiplicative under the tensor product of vector bundles: $ch(\xi\otimes\eta)=ch(\chi)ch(\eta)$. Therefore we get a ring homomorphism.

\hypertarget{for_spectra_and_generalized_cohomology_theories}{}\subsubsection*{{For spectra and generalized cohomology theories}}\label{for_spectra_and_generalized_cohomology_theories}

Let $E$ be a [[spectrum]] The [[isomorphism]]

\begin{displaymath}
H(-,E)\otimes \mathbb{R}
  \;\;
  \stackrel{\simeq}{\to}
  \;\;
  H(-,(\pi_* E)\otimes \mathbb{R})
\end{displaymath}
that defines the Chern-character map is induced by a canonical cocycle on the [[spectrum]] $E$ that is called the \textbf{fundamental cocycle}.

This is described for instance in section \href{http://arxiv.org/PS_cache/math/pdf/0211/0211216v2.pdf#page=47}{4.8, page 47} of Hopkins-Singer [[Quadratic Functions in Geometry, Topology,and M-Theory]].

\hypertarget{for_cohesive_stable_homotopy_types}{}\subsubsection*{{For cohesive stable homotopy types}}\label{for_cohesive_stable_homotopy_types}

More generally, for $\hat E$ a [[stable homotopy type]] in a [[cohesive (∞,1)-topos]], then the underlying bare homotopy type is $E \coloneqq \Pi(\hat E)$ and the corresponding Chern character is

\begin{displaymath}
ch \coloneqq \Pi \theta_{\hat E} \;\colon\; E \simeq \Pi(\hat E) \longrightarrow \Pi \flat_{dR} \hat E
  \,.
\end{displaymath}
For more on this see at \emph{[[differential cohomology diagram]]}.

\hypertarget{in_terms_of_cyclic_homology}{}\subsubsection*{{In terms of cyclic homology}}\label{in_terms_of_cyclic_homology}

Generalizing in another direction, generalized Chern characters are given by passage to [[derived loop spaces]] and their [[cyclic homology]] or, more generally, [[topological cyclic homology]] (\hyperlink{ToenVezzosi08}{Toen-Vezzosi 08}, \hyperlink{HoyoisScherotzkeSibillia15}{Hoyois-Scherotzke-Sibillia 15}).

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

\hypertarget{pushforward_and_grothendieckriemannroch_theorem}{}\subsubsection*{{Push-forward and Grothendieck-Riemann-Roch theorem}}\label{pushforward_and_grothendieckriemannroch_theorem}

The behaviour of the Chern-character under [[fiber integration in generalized cohomology]] along [[proper maps]] is described by the [[Grothendieck-Riemann-Roch theorem]].

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

\begin{itemize}%
\item [[odd Chern character]]


\item for [[algebraic K-theory]]:

\begin{itemize}%
\item [[cyclotomic trace]]


\item [[regulator]], [[Beilinson regulator]]



\end{itemize}

\item [[Grothendieck-Riemann-Roch theorem]]


\item [[higher chromatic Chern character]]

\begin{itemize}%
\item [[elliptic Chern character]]


\item [[Morava E-theoretic Chern character]]



\end{itemize}

\item [[transchromatic character]]


\item [[group character]]



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

The universal Chern character for [[generalized (Eilenberg-Steenrod) cohomology theory]] is discussed in section \href{http://arxiv.org/PS_cache/math/pdf/0211/0211216v2.pdf#page=47}{4.8, page 47} of

\begin{itemize}%
\item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology,and M-Theory]]}, (\href{http://arxiv.org/abs/math.AT/0211216}{math.AT/0211216}).

\end{itemize}
in the context of [[differential cohomology]] via [[differential function complexes]].

The observation putting this into the general context of [[differential cohomology diagrams]] (see there) of [[stable homotopy types]] in [[cohesion]] is due to

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], section 4.4. of \emph{Differential cohomology theories as sheaves of spectra} (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188})

\end{itemize}
based on

\begin{itemize}%
\item [[Ulrich Bunke]], [[David Gepner]], around def, 2.1 of \emph{Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory} (\href{http://arxiv.org/abs/1306.0247}{arXiv:1306.0247})

\end{itemize}
A characterization of Chern-character maps for [[K-theory]] is in

\begin{itemize}%
\item [[Goncalo Tabuada]], \emph{A universal characterization of the Chern character maps} (\href{http://arxiv.org/abs/1002.3726}{arXiv/1002.3276})

\end{itemize}
Discussion for [[equivariant K-theory]]:

\begin{itemize}%
\item German Stefanich, \emph{Chern Character in Twisted and Equivariant K-Theory} (\href{https://math.berkeley.edu/~germans/Chern2.pdf}{pdf})

\end{itemize}
A discussion of Chern characters in terms of [[free loop space objects]] in [[derived geometry]] is in

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{A note on Chern character, loop spaces and derived algebraic geometry} (\href{http://arxiv.org/abs/0804.1274}{arXiv:0804.1274})

\end{itemize}
which conjectures a construction that is fully developed in

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{Caract\`e{}res de Chern, traces \'e{}quivariantes et g\'e{}om\'e{}trie alg\'e{}brique d\'e{}riv\'e{}e} (\href{http://arxiv.org/abs/0903.3292}{arXiv:0903.3292})

\end{itemize}
See also

\begin{itemize}%
\item [[Marc Hoyois]], \emph{Chern character and derived algebraic geometry} (2009) (\href{http://math.mit.edu/~hoyois/papers/chern.pdf}{pdf})


\item [[Marc Hoyois]], [[Sarah Scherotzke]], [[Nicolò Sibilla]], \emph{Higher traces, noncommutative motives, and the categorified Chern character} (\href{http://arxiv.org/abs/1511.03589}{arXiv:1511.03589})



\end{itemize}
[[!redirects Chern characters]]

[[!redirects Chern character map]] [[!redirects Chern character maps]]



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