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\section*{KR cohomology theory}

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

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

[[!include cohomology - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation 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{as_the_grothendieck_group_of_complex_vector_bundles_with_real_structure}{As the Grothendieck group of complex vector bundles with real structure}\dotfill \pageref*{as_the_grothendieck_group_of_complex_vector_bundles_with_real_structure} \linebreak
\noindent\hyperlink{as_a_genuine_spectrum}{As a genuine $\mathbb{Z}_2$-Spectrum}\dotfill \pageref*{as_a_genuine_spectrum} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{bigrading}{Bigrading}\dotfill \pageref*{bigrading} \linebreak
\noindent\hyperlink{InducedFromModuliStackOfTori}{As induced from the derived moduli stack of tori}\dotfill \pageref*{InducedFromModuliStackOfTori} \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}

What is called \emph{KR-theory} (\hyperlink{Atiyah66}{Atiyah 66}) is variant of [[topological K-theory]] on [[spaces]] equipped with a $\mathbb{Z}_2$-[[action]] (by [[homeomorphism]], hence equipped with one [[involution|involutive]] homeomorphism -- a ``[[real space]]'').

In terms of [[cocycle]] models, classes of KR-theory are represented by [[complex vector bundles]] over $X$ on which the involution on their base space lifts to an \emph{anti-linear} involution of the total space. Over manifolds with trivial involution these are precisely the [[complexification]] of [[real vector bundles]] and hence over such spaces $KR$-theory reduces to [[KO]]-[[cohomology theory|theory]]. Conversely, over two copies $X \cup X$ of $X$ equipped with the involution that interchanges the two, $KR$-theory reduces to [[KU]]-[[cohomology theory|theory]]. Finally over $X \times S^1$ with the involution the antipodal identification on the second ([[circle]]) factor , $KR$-theory reduces to the self-conjugate [[KSC-theory]] (\hyperlink{Anderson64}{Anderson 64}). So in general $KR$-theory interpolates between all these cases. For instance on $X \times S^1$ with the reflection-involution on the circle (the [[real space]] denoted $S^{1,1}$, the non-trivial $\mathbb{Z}_2$-[[representation sphere]]) it behaves like $KO$-theory at the two involution fixed points (the two [[O-planes]]) and like $KU$ in their complement (a model that makes this very explicit is given in \hyperlink{DMR13}{DMR 13, section 4}), schematically:

\begin{displaymath}
KR(S^{1,1}) = ( KO --- KU --- KO )
\end{displaymath}
More abstractly, complex conjugation of complex vector bundles induces on the [[complex K-theory]] [[spectrum]] [[KU]] an involutive [[automorphism]]. $KR$ is the corresponding $\mathbb{Z}_2$-[[equivariant spectrum]], and $KR$-theory the corresponding $\mathbb{Z}_2$-[[equivariant cohomology]] theory. In particular, the [[homotopy fixed point]] of [[KU]] under this automorphism is [[KO]]

\begin{displaymath}
KO \simeq (KU)^{\mathbb{Z}/2}
\end{displaymath}
(e.g.\hyperlink{Karoubi01}{Karoubi 01}, \hyperlink{Dugger03}{Dugger 03, corollary 7.6}, \hyperlink{HillHopkinsRavenel}{Hill-Hopkins-Ravenel, section 7.3}) and this way where in complex K-theory one has [[KU]]-[[modules]] ([[∞-modules]]), so in KR-theory one has $KO$-modules.

KR is an example of a [[real-oriented cohomology theory]], together with for instance [[MR-theory]] and [[BPR-theory]].

\begin{remark}
\label{}\hypertarget{}{}
An [[involution]] on a space by a [[homeomorphism]] (or [[diffeomorphism]]) as they appear in KR theory may be thought of as a ``non-linear [[real structure]]'', and therefore spaces equipped with such involutions are called ``[[real spaces]]''. Following this, $KR$-theory is usually pronounced ``real K-theory''. But \textbf{beware} that this terminology easily conflicts with or is confused with [[KO]]-theory. For disambiguation the latter might better be called ``orthogonal K-theory''. But on abstract grounds maybe $KR$-theory would best be just called $\mathbb{Z}_2$-equivariant complex K-theory.

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

\hypertarget{as_the_grothendieck_group_of_complex_vector_bundles_with_real_structure}{}\subsubsection*{{As the Grothendieck group of complex vector bundles with real structure}}\label{as_the_grothendieck_group_of_complex_vector_bundles_with_real_structure}

\ldots{}(\hyperlink{Atiyah66}{Atiyah 66})\ldots{}

\hypertarget{as_a_genuine_spectrum}{}\subsubsection*{{As a genuine $\mathbb{Z}_2$-Spectrum}}\label{as_a_genuine_spectrum}

The following gives $KR$ as a [[genuine G-spectrum]] for $G = \mathbb{Z}_2$.

Using that every orthogonal representation of $\mathbb{Z}_2$ is contained in an $\mathbb{C}^n$ with its [[complex conjugation]] action, one can restrict attention to these. Write

\begin{displaymath}
\mathbb{C}P^1 = S^{2,1} = S^{\mathbb{C}}
  \,.
\end{displaymath}
The reduced [[canonical line bundle]] over this (the [[Hopf fibration]]) is classified by a map

\begin{displaymath}
S^{2,1}= \mathbb{C}P^1 \to \mathbb{Z}\times BU
\end{displaymath}
to the [[classifying space]] for [[topological K-theory]]. The homotopy-associative multiplication on this space then yields the structure map of a $\mathbb{Z}_2$-spectrum

\begin{displaymath}
S^{2,1} \wedge (\mathbb{Z} \times BU)\to \mathbb{Z}\times BU
  \,.
\end{displaymath}
This is in fact an [[Omega spectrum]], by equivariant complex [[Bott periodicity]] (for instance in \hyperlink{Dugger03}{Dugger 03, p. 2-3}).

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

\hypertarget{bigrading}{}\subsubsection*{{Bigrading}}\label{bigrading}

As any genuine [[equivariant cohomology theory]] $KR$-theory is naturally graded over the [[representation ring]] $RO(\mathbb{Z}_2)$. Write $\mathbb{R}$ for the trivial 1-dimensional representation and $\mathbb{R}_-$ for that given by the sign involution. Then the general orthogonalrepresentation decomposes as a [[direct sum]]

\begin{displaymath}
V = \mathbb{R}^+\oplus \mathbb{R}_-^q
  \,.
\end{displaymath}
The corresponding [[representation sphere]] is

\begin{displaymath}
S^V = (some\; convention)
  \,.
\end{displaymath}
\hypertarget{InducedFromModuliStackOfTori}{}\subsubsection*{{As induced from the derived moduli stack of tori}}\label{InducedFromModuliStackOfTori}

The relation between $KU$, $KO$ and $KR$ naturally arises in [[chromatic homotopy theory]] as follows.

Inside the [[moduli stack of formal group laws]] sits the \emph{[[moduli stack of tori|moduli stack of one dimensional tori]]} $\mathcal{M}_{\mathbb{G}_m}$ (\hyperlink{LawsonNaumann12}{Lawson-Naumann 12, def. A.1, A.3}). This is equivalent to the [[quotient stack]] of the point by the [[group of order 2]]

\begin{displaymath}
\mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2
\end{displaymath}
(\hyperlink{LawsonNaumann12}{Lawson-Naumann 12, prop. A.4}). Here the $\mathbb{Z}_2$-action is the inversion involution on [[abelian groups]].

Using the [[Goerss-Hopkins-Miller theorem]] this stack carries an [[E-∞ ring]]-valued [[structure sheaf]] $\mathcal{O}^{top}$ (\hyperlink{LawsonNaumann12}{Lawson-Naumann 12, theorem A.5}); and by the above equivalence this is a single [[E-∞ ring]] equipped with a $\mathbb{Z}_2$-[[∞-action]]. This is [[KU]] with its involution induced by [[complex conjugation]], hence essentially is $KR$.

Accordingly, the [[global sections]] of $\mathcal{O}^{top}$ over $\mathcal{M}_{\mathbb{G}_m}$ are the $\mathbb{Z}_2$-[[homotopy fixed points]] of this action, hence is $KO$. This is further amplified in (\hyperlink{Mathew13}{Mathew 13, section 3}) and (\hyperlink{Mathew}{Mathew, section 2}).

As suggested there and by the main (\hyperlink{LawsonNaumann12}{Lawson-Naumann 12, theorem 1.2}) this realizes (at least localized at $p = 2$) the inclusion $KO \to KU$ as the restriction of an analogous inclusion of [[tmf]] built as the global sections of the similarly derived [[moduli stack of elliptic curves]].

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

\begin{itemize}%
\item [[real-oriented cohomology theory]]

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


\item [[BPR-theory]]



\end{itemize}

\item [[real algebraic K-theory]]


\item [[orientifold]]


\item [[KO-dimension]]



\end{itemize}
[[!include chromatic tower examples - table]]

[[!include string theory and cohomology theory -- table]]

[[!include moduli stack of curves -- table]]

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

KR theory was introduced in

\begin{itemize}%
\item [[Michael Atiyah]], \emph{K-theory and reality}, The Quarterly Journal of Mathematics. Oxford. Second Series 17 (1) (1966),: 367--386, ISSN 0033-5606, MR 0206940 (\href{https://doi.org/10.1093/qmath/17.1.367}{doi:10.1093/qmath/17.1.367}, [[AtiyahKReal.pdf:file]])

\end{itemize}
The version of $KSC$-theory was introduced in

\begin{itemize}%
\item D. W. Anderson, \emph{The real K-theory of classifying spaces} Proc. Nat. Acad. Sci. U. S. A., 51(4):634--636, 1964.

\end{itemize}
The dual concept of KR-homology was defined in

\begin{itemize}%
\item [[Gennady Kasparov]], \emph{The operator K-functor and extensions of $C^\ast$-algebras,} Izv. Akad. Nauk. SSSR Ser. Mat. 44, 571-636 (1980).

\end{itemize}
Computations over [[compact Lie groups]] are spelled out in

\begin{itemize}%
\item Chi-Kwong Fok, \emph{The real K-Theory of compact Lie groups}, 2014 (\href{http://pi.math.cornell.edu/~ckfok/Chi_Kwong_Fok_thesis.pdf}{pdf})

\end{itemize}
Discussion in the general context of [[real oriented cohomology theory]] is in

\begin{itemize}%
\item Po Hu, [[Igor Kriz]], \emph{Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence}, Topology 40 (2001) 317-399 (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf}{pdf})

\end{itemize}
Further discussion includes

\begin{itemize}%
\item [[Johan Dupont]], \emph{Symplectic bundles and $KR$-theory} (1967) (\href{http://www.mscand.dk/article.php?id=1908}{pdf})

\end{itemize}
Reviews include

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/KR-theory}{KR-theory}}


\item Paolo Masulli, \emph{Equivariant homotopy: $KR$-theory}, Master thesis (2011) (\href{http://www.math.ku.dk/~jg/students/masulli.msthesis.2011.pdf}{pdf})



\end{itemize}
Remarks on homotopy-theoretic KR in the context of [[algebraic K-theory]] are in

\begin{itemize}%
\item [[Vigleik Angeltveit]], [[Andrew Blumberg]], Teena Gerhardt, [[Michael Hill]], \emph{Algebraic K-theory and equivariant homotopy theory}, 2012 (\href{http://www.birs.ca/workshops/2012/12w5116/report12w5116.pdf}{pdf})

\end{itemize}
Discussion of [[equivariant K-theory|equivariant]] and [[twisted K-theory|twisted]] versions of KR-theory

\begin{itemize}%
\item [[El-kaïoum M. Moutuou]], \emph{Twistings of KR for Real groupoids} (\href{http://arxiv.org/abs/1110.6836}{arXiv:1110.6836})


\item [[El-kaïoum M. Moutuou]], \emph{Graded Brauer groups of a groupoid with involution}, J. Funct. Anal. 266 (2014), no.5 (\href{https://arxiv.org/abs/1202.2057}{arXiv:1202.2057})


\item [[Daniel Freed]], \emph{Lectures on twisted K-theory and orientifolds}, lectures at ESI Vienna, 2012 ([[FreedESI2012.pdf:file]])


\item [[Daniel Freed]], [[Gregory Moore]], Section 7 of: \emph{Twisted equivariant matter}, Ann. Henri Poincaré (2013) 14: 1927 (\href{https://arxiv.org/abs/1208.5055}{arXiv:1208.5055})


\item [[Kiyonori Gomi]], \emph{Freed-Moore K-theory} (\href{https://arxiv.org/abs/1705.09134}{arXiv:1705.09134}, \href{http://inspirehep.net/record/1601772}{spire:1601772})



\end{itemize}
This is with motivation from \emph{[[orientifolds]]}, see the references given there for more. A long list of computations of twisted $KR$-classes on tori with applications to [[T-duality]] on [[orientifolds]]/[[O-planes]] is in

\begin{itemize}%
\item [[Sergei Gukov]], \emph{K-Theory, Reality, and Orientifolds}, Commun.Math.Phys. 210 (2000) 621-639 (\href{http://arxiv.org/abs/hep-th/9901042}{arXiv:hep-th/9901042})


\item [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], \emph{T-duality For Orientifolds and Twisted KR-theory} (\href{http://arxiv.org/abs/1306.1779}{arXiv:1306.1779})

(but see \hyperlink{HMSV19}{HMSV 19, p.5 footnote 1})


\item [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], \emph{String theory on elliptic curve orientifolds and KR-theory} (\href{http://arxiv.org/abs/1402.4885}{arXiv:1402.4885})



\end{itemize}
A general proposal for [[differential K-theory|differential]] [[equivariant K-theory|equivariant]] KR-theory of [[orientifolds]] and [[O-plane charge]]

\begin{itemize}%
\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides})

\end{itemize}
Discussion of $KO$ as the $\mathbb{Z}_2$-[[homotopy fixed points]] of $KU$ (or $KR$) is in

\begin{itemize}%
\item [[Max Karoubi]], \emph{A descent theorem in topological K-theory}, K-theory 24 (2001), no. 2, 109--114 (\href{http://arxiv.org/abs/math/0509396}{arXiv:math/0509396})


\item [[Daniel Dugger]], \emph{An Atiyah-Hirzebruch spectral sequence for $KR$-theory}, Ktheory 35 (2005), 213--256. (\href{http://arxiv.org/abs/math/0304099}{arXiv:0304099})


\item [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], section 7.3 of \emph{The Arf-Kervaire problem in algebraic topology: Sketch of the proof} ([[HHRKervaire.pdf:file]])



\end{itemize}
Discussion of $KU$ with its $\mathbb{Z}_2$-action as the [[E-∞ ring]]-valued [[structure sheaf]] of the moduli stack of tori is due to

\begin{itemize}%
\item [[Tyler Lawson]], [[Niko Naumann]], appendix of \emph{Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2} (\href{http://arxiv.org/abs/1203.1696}{arXiv:1203.1696})

\end{itemize}
which is reviewed and amplified further in

\begin{itemize}%
\item [[Akhil Mathew]], section 3 of \emph{The homology of $tmf$} (\href{http://arxiv.org/abs/1305.6100}{arXiv:1305.6100})


\item Akhil Mathew, section 2 of \emph{The homotopy groups of $TMF$}, talk notes (\href{http://math.mit.edu/~sglasman/tmfhomotopy.pdf}{pdf})



\end{itemize}
Discussion of [[twisted cohomology|twists]] of [[KR-theory]] by [[HZR-theory]] in degree 3 via [[bundle gerbes]] ([[Jandl gerbes]]) suitable for classifying [[D-brane charge]] on [[orientifolds]]:

\begin{itemize}%
\item [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], \emph{Real bundle gerbes, orientifolds and twisted KR-homology} (\href{http://arxiv.org/abs/1608.06466}{arXiv:1608.06466})


\item [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], \emph{Sign choices for orientifolds} (\href{https://arxiv.org/abs/1905.06041}{arXiv:1905.06041})



\end{itemize}
[[!redirects KR-homology]]

[[!redirects KR-theory]]

[[!redirects KR]]

[[!redirects real K-theory]]



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