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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{twisted K-theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - 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{by_sections_of_associated_bundles}{By sections of associated $K U$-bundles}\dotfill \pageref*{by_sections_of_associated_bundles} \linebreak \noindent\hyperlink{ByBundlesOfFredholmOperators}{By bundles of Fredholm operators}\dotfill \pageref*{ByBundlesOfFredholmOperators} \linebreak \noindent\hyperlink{by_twisted_vector_bundles_gerbe_modules}{By twisted vector bundles (gerbe modules)}\dotfill \pageref*{by_twisted_vector_bundles_gerbe_modules} \linebreak \noindent\hyperlink{by_kktheory_of_twisted_convolution_algebras}{By KK-theory of twisted convolution algebras}\dotfill \pageref*{by_kktheory_of_twisted_convolution_algebras} \linebreak \noindent\hyperlink{other_constructions}{Other constructions}\dotfill \pageref*{other_constructions} \linebreak \noindent\hyperlink{twists}{Twists}\dotfill \pageref*{twists} \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} \emph{Twisted K-theory} is a [[twisted cohomology]] version of ([[topological K-theory|topological]]) [[K-theory]]. The most famous twist is by a class in degree 3 [[ordinary cohomology]] (geometrically a $U(1)$-[[bundle gerbe]] or [[circle 2-group]]-[[principal 2-bundle]]), but there are various other twists. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{by_sections_of_associated_bundles}{}\subsubsection*{{By sections of associated $K U$-bundles}}\label{by_sections_of_associated_bundles} Write [[KU]] for the [[spectrum]] of complex [[topological K-theory]]. Its degree-0 space is, up to [[weak homotopy equivalence]], the space \begin{displaymath} B U \times \mathbb{Z} = {\lim_\to}_n B U(n) \times \mathbb{Z} \end{displaymath} or the space $Fred(\mathcal{H})$ of [[Fredholm operator]]s on some separable [[Hilbert space]] $\mathcal{H}$. \begin{displaymath} (K U)_0 \simeq B U \times \mathbb{Z} \simeq Fred(\mathcal{H}) \,. \end{displaymath} The ordinary [[topological K-theory]] of a suitable [[topological space]] $X$ is given by the set of [[homotopy classes]] of maps from (the [[suspension spectrum]] of) $X$ to $KU$: \begin{displaymath} K(X)_\bullet \simeq [X, (K U)_\bullet] \,. \end{displaymath} The [[projective unitary group]] $P U(\mathcal{H})$ (a [[topological group]]) acts canonically by [[automorphism]]s on $(K U)_0$. (This follows by the identificatioon of $KU_0$ with the space of [[Fredholm operators]], see \hyperlink{ByBundlesOfFredholmOperators}{below}) Therefore for $P \to X$ any $PU(\mathcal{H})$-[[principal bundle]], we can form the [[associated bundle]] $P \times_{P U(\mathcal{H})} (K U)_0$. Since the [[homotopy type]] of $P U(\mathcal{H})$ is that of an [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$, there is precisely one isomorphism class of such bundles representing a class $\alpha \in H^3(X, \mathbb{Z})$. \begin{defn} \label{SpectrumBundDefinition}\hypertarget{SpectrumBundDefinition}{} The \textbf{twisted K-theory} with twist $\alpha \in H^3(X, \mathbb{Z})$ is the set of [[homotopy]]-classes of [[section]]s of such a bundle \begin{displaymath} K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} (K U)_0) \,. \end{displaymath} Similarily the reduced $\alpha$-twisted K-theory is the subset \begin{displaymath} \tilde K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} B U) \,. \end{displaymath} a \end{defn} \hypertarget{ByBundlesOfFredholmOperators}{}\subsubsection*{{By bundles of Fredholm operators}}\label{ByBundlesOfFredholmOperators} The following is due to (\hyperlink{AtiyahSinger69}{Atiyah-Singer 69}, \hyperlink{AtiyahSegal04}{Atiyah-Segal 04}). Write \begin{itemize}% \item $Cl_n \coloneqq Cl^{\mathbb{C}}(\mathbb{R}^n,\langle -,-\rangle)$ for the [[complexification]] of the [[Clifford algebra]] of the [[Cartesian space]] $\mathbb{R}^n$ with its standard [[inner product]]; \item $S_n$ for its $\mathbb{Z}/2\mathbb{Z}$-[[graded module|graded]] [[irreducible module]] (see at \emph{[[spin representation]]}); \item $H_0$ for [[generalized the|the]] $\mathbb{Z}/2\mathbb{Z}$-graded [[separable Hilbert space]] whose even and odd part are both infinite-dimensional. \end{itemize} \begin{defn} \label{Fredn}\hypertarget{Fredn}{} For $n \in \mathbb{N}$, the [[topological space]] $Fred^{(n)}$ of [[Fredholm operators]] on $S_n \otimes H_0$ is the [[set]] \begin{displaymath} Fred^{(n)} \coloneqq \left\{ F \in \mathcal{B}(S_n \otimes H_0) \;|\; F \, odd\,, F^\ast = F \,, F^2 - 1 \in \mathcal{K}(S_n \otimes H_0)\,, [F,\gamma] = 0 \, for\, \gamma \in Cl_n \right\} \end{displaymath} (where $\mathcal{B}$ denotes [[bounded operators]] and $\mathcal{K}$ denotes [[compact operators]] and where $[-,-]$ denotes the [[graded commutator]]) and the [[topological space|topology]] on this set is the [[subspace topology]] induced by the embedding \begin{displaymath} Fred^{(n)} \hookrightarrow \mathcal{B}(S_n \otimes H_0) \times \mathcal{K}(S_n \otimes H_0) \end{displaymath} given by \begin{displaymath} F\mapsto (F, F^2 - 1) \,, \end{displaymath} where $\mathcal{B}$ is equipped with the [[compact-open topology]] and $\mathcal{K}$ with the [[norm topology]]. \end{defn} (\hyperlink{AtiyahSinger69}{Atiyah-Singer 69, p. 7}, \hyperlink{AtiyahSegal04}{Atiyah-Segal 04, p. 21}, \hyperlink{FreedHopkinsTeleman11}{Freed-Hopkins-Teleman 11, def. A.40}) These spaces indeed form a model for the [[KU]] [[spectrum]]: \begin{prop} \label{}\hypertarget{}{} For all $n \in \mathbb{N}$ there are [[natural equivalence|natural]] [[weak homotopy equivalences]] \begin{displaymath} Fred^{(n+1)} \stackrel{\simeq}{\longrightarrow} \Omega Fred^{(n)} \end{displaymath} and \begin{displaymath} Fred^{(n+2)} \stackrel{\simeq}{\longrightarrow} Fred^{(n)} \end{displaymath} between the spaces of graded [[Fredholm operators]] of def. \ref{Fredn} and their [[loop spaces]]. \end{prop} (\hyperlink{AtiyahSinger69}{Atiyah-Singer 69, theorem B(k)}, \hyperlink{AtiyahSegal04}{Atiyah-Segal 04 (4.2)}, \hyperlink{FreedHopkinsTeleman11}{Freed-Hopkins-Teleman 11, below def. A.40}) Regard the [[stable unitary group]] $U(H_0)$ as equipped with the [[subspace topology]] induced by the inclusion \begin{displaymath} U(H_0) \stackrel{(id,(-)^{-1})}{\hookrightarrow} \mathcal{B}(H_0)\times\mathcal{B}(H_0) \end{displaymath} from the [[compact-open topology]] on the [[bounded linear operators]]. \begin{prop} \label{}\hypertarget{}{} The [[conjugation action]] of the [[stable unitary group]] $U(H_0)$ on $Fred^{(n)}$, def. \ref{Fredn}, is [[continuous functions|continuous]]. \end{prop} This follows with (\hyperlink{AtiyahSegal04}{Atiyah-Segal 04, prop. A1.1}). \begin{defn} \label{}\hypertarget{}{} Given a class $\chi \in H^3(X,\mathbb{Z})$ represented by a $PU(H_0)$-bundle $P \to X$ with [[associated bundle|associated]] Fredholm bundle \begin{displaymath} Fred^{(n)+ \chi} \coloneqq P \underset{PU(H_0)}{\times} Fred^{(n)} \,, \end{displaymath} then the corresponding $\chi$-twisted [[cohomology]] [[spectrum]] is that consisting of the [[spaces of sections]] \begin{displaymath} \Gamma(X, Fred^{(n)+ \chi}) \,. \end{displaymath} \end{defn} (\hyperlink{FreedHopkinsTeleman11}{Freed-Hopkins-Teleman 11, def. 3.14}) \hypertarget{by_twisted_vector_bundles_gerbe_modules}{}\subsubsection*{{By twisted vector bundles (gerbe modules)}}\label{by_twisted_vector_bundles_gerbe_modules} \begin{defn} \label{TwBundDefinition}\hypertarget{TwBundDefinition}{} Let $\alpha \in H^3(X, \mathbb{Z})$ be a class in degree-3 [[integral cohomology]] and let $P \in \mathbf{H}^3(X, \mathbf{B}^2 U(1))$ be any [[cocycle]] representative, which we may think of either as giving a [[circle n-bundle with connection|circle 2-bundle]] or a [[bundle gerbe]]. Write $TwBund(X, P)$ for the [[groupoid]] of [[twisted bundle]]s on $X$ with twist given by $P$. Then let \begin{displaymath} \tilde K_\alpha(X) := TwBund(X,P) \end{displaymath} be the set of [[isomorphism]] classes of twisted bundles. Call this the \textbf{twisted K-theory} of $X$ with twist $\alpha$. \end{defn} \begin{quote}% (Some technical details need to be added for the non-torsion case.) \end{quote} \begin{prop} \label{}\hypertarget{}{} This definition of twisted $K_0$ is equivalent to that of prop. \ref{SpectrumBundDefinition}. \end{prop} This is (\hyperlink{CBMMS}{CBMMS, prop. 6.4, prop. 7.3}). \hypertarget{by_kktheory_of_twisted_convolution_algebras}{}\subsubsection*{{By KK-theory of twisted convolution algebras}}\label{by_kktheory_of_twisted_convolution_algebras} A [[circle 2-group]] [[principal 2-bundle]] is also incarnated as a [[centrally extended Lie groupoid]]. The corresponding [[twisted groupoid convolution algebra]] has as its [[operator K-theory]] the twisted K-theory of the base space (or base-[[stack]]). See at \emph{[[KK-theory]]} for more on this. \hypertarget{other_constructions}{}\subsection*{{Other constructions}}\label{other_constructions} Let $Vectr$ be the [[stack]] of [[vectorial bundle]]s. (If we just take [[vector bundle]]s we get a notion of twisted K-theory that only allows twists that are finite order elements in their [[cohomology group]]). There is a canonical morphism \begin{displaymath} \rho : \mathbf{B} U(1) \to Vect \hookrightarrow Vectr \end{displaymath} coming from the standard [[representation]] of the [[group]] $U(1)$. Let $\mathbf{B}_{\otimes} Vectr$ be the [[delooping]] of $Vectr$ with respect to the [[tensor product]] [[monoidal category|monoidal structure]] (not the additive structure). Then we have a [[fibration sequence]] \begin{displaymath} Vectr \to {*} \to \mathbf{B}_\otimes Vectr \end{displaymath} of [[(infinity,1)-category|(infinity,1)-categories]] (instead of [[infinity-groupoid]]s). The entire morphism above deloops \begin{displaymath} \mathbf{B}\rho : \mathbf{B}^2 U(1) \to \mathbf{B}_\otimes Vect \hookrightarrow \mathbf{B}_{\otimes} Vectr \end{displaymath} being the standard representation of the [[2-group]] $\mathbf{B}U(1)$. From the general nonsense of [[twisted cohomology]] this induces canonically now for every $\mathbf{B}^2 U(1)$-[[cohomology|cocycle]] $c$ (for instance given by a [[bundle gerbe]]) a notion of $c$-twisted $Vectr$-cohomology: \begin{displaymath} \itexarray{ \mathbf{H}^c(X, Vectr) &\to& {*} \\ \downarrow && \downarrow^{\mathbf{B}\rho \circ c} \\ {*} &\to& \mathbf{H}(X,\mathbf{B}_\otimes Vectr) } \,. \end{displaymath} After unwrapping what this means, the result of (\hyperlink{Gomi}{Gomi}) shows that [[concordance]] classes in $\mathbf{H}^c(X,Vectr)$ yield twisted K-theory. \hypertarget{twists}{}\subsection*{{Twists}}\label{twists} By the general discussion of [[twisted cohomology]] the [[moduli space]] for the twists of [[periodic complex K-theory]] $KU$ is the [[Picard ∞-group]] in $KU Mod$. The ``geometric'' twists among these have as moduli space the non-connected delooping $bgl_1^\ast(KU)$ of the [[∞-group of units]] of $KU$. A model for this in 4-truncation is given by [[super line 2-bundles]]. For the moment see there for further discussion and further references. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[K-theory]] \item [[topological K-theory]], [[KR-theory]] \item \textbf{twisted K-theory} \begin{itemize}% \item [[differential K-theory]] \item [[twisted differential K-theory]] \item [[fiber integration in K-theory]] \end{itemize} \item [[twisted ordinary cohomology]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} A textbook account is in \begin{itemize}% \item [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], \emph{[[Basic Bundle Theory and K-Cohomology Invariants]]}, Lecture Notes in Physics, Springer 2008 (\href{http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf}{pdf}) \end{itemize} The concept of twisted K-theory originates in \begin{itemize}% \item [[Max Karoubi]], \emph{Alg\`e{}bres de Clifford et K-th\'e{}orie.} Ann. Sci. Ecole Norm. Sup. (4), pp. 161-270 (1968). \item [[Peter Donovan]], [[Max Karoubi]], \emph{Graded Brauer groups and $K$-theory with local coefficients}, Publications Math\'e{}matiques de l'IH\'E{}S, 38 (1970), p. 5-25 (\href{http://www.numdam.org/item?id=PMIHES_1970__38__5_0}{numdam}) \end{itemize} which discusses twists of $KO$ and $KU$ over some $X$ by elements in $H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})$. The formulation in terms of sections of Fredholm bundles seems to go back to \begin{itemize}% \item [[Jonathan Rosenberg]], \emph{Continuous-trace algebras from the bundle theoretic point of view} , J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368-381. \end{itemize} A comprehensive account of twisted K-theory with twists in $H^3(X, \mathbb{Z})$ is in \begin{itemize}% \item [[Michael Atiyah]], [[Isadore Singer]], \emph{Index theory for skew-adjoint Fredholm operators}, Publications Math\'e{}matiques de l'Institut des Hautes \'E{}tudes Scientifiques January 1969, Volume 37, Issue 1, pp 5-26 (\href{http://www.maths.ed.ac.uk/~aar/papers/askew.pdf}{pdf}) \item [[Michael Atiyah]], [[Graeme Segal]], \emph{Twisted K-theory} (\href{http://arxiv.org/abs/math/0407054}{arXiv:math/0407054}) \item [[Michael Atiyah]], [[Graeme Segal]], \emph{Twisted K-theory and cohomology} (\href{http://arxiv.org/abs/math/0510674}{arXiv:math/0510674}) \end{itemize} and for more general twists in \begin{itemize}% \item [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], \emph{[[Loop Groups and Twisted K-Theory]] I} , J. Topology, 4 (2011), 737-789 (\href{http://arxiv.org/abs/0711.1906}{arXiv:0711.1906}) \end{itemize} The seminal result on the relation to [[loop group]] [[representations]], now again with twists in $H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})$, is in the series of articles \begin{itemize}% \item [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], \emph{Twisted K-theory and loop group representations} \href{http://arxiv.org/abs/math/0312155}{arXiv:math/0312155}; \emph{[[Loop Groups and Twisted K-Theory]] I} (\href{http://arxiv.org/abs/0711.1906}{arXiv:0711.1906}) ; \emph{[[Loop Groups and Twisted K-Theory]] II} (\href{http://arxiv.org/abs/math/0511232}{arXiv:math/0511232}) \end{itemize} The result on twisted K-groups has been lifted to an equivalence of categories in \begin{itemize}% \item [[Daniel Freed]], [[Constantin Teleman]], \emph{Dirac families for loop groups as matrix factorizations}, \href{http://arxiv.org/abs/1409.6051}{arxiv/1409.6051} \end{itemize} Discussion in terms of [[Karoubi K-theory]]/[[Clifford module bundles]] is in \begin{itemize}% \item [[Max Karoubi]], \emph{Clifford modules and twisted K-theory}, Proceedings of the International Conference on Clifford algebras (ICCA7) (\href{http://arxiv.org/abs/0801.2794}{arXiv:0801.2794}) \end{itemize} The perspective of twisted K-theory by sections of a $K U$-bundle of spectra ([[parameterized spectra]]) is discussed in \begin{itemize}% \item [[Peter May|May]], Sigurdsson, section 22 of \emph{Parametrized homotopy theory} (\href{http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf}{pdf}) AMS Lecture notes 132 \item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], section 7 of \emph{Twists of K-theory and TMF}, in [[Jonathan Rosenberg]] et al. (eds.), \emph{Superstrings, Geometry, Topology, and $C^\ast$-algebras}, volume 81 of \emph{Proceedings of Symposia in Pure Mathematics}, 2009 (\href{http://arxiv.org/abs/1002.3004}{arXiv:1002.3004}) \end{itemize} See the references at \emph{[[(infinity,1)-vector bundle]]} for more on this. Discussion in terms of [[twisted bundles]]/[[bundle gerbe modules]] is in \begin{itemize}% \item [[Peter Bouwknegt]], [[Alan Carey]], [[Varghese Mathai]], [[Michael Murray]] and [[Danny Stevenson]], \emph{K-theory of bundle gerbes and twisted K-theory} , Commun Math Phys, 228 (2002) 17-49 (\href{http://arxiv.org/abs/hep-th/0106194}{arXiv:hep-th/0106194}) \end{itemize} but apparently contains a mistake, as pointed out in \begin{itemize}% \item [[Alan Carey]], [[Bai-Ling Wang]], top of p. 10 in \emph{Thom isomorphism and Push-forward map in twisted K-theory} (\href{https://arxiv.org/abs/math/0507414}{arXiv:math/0507414}) \end{itemize} The generalization of this to [[groupoid K-theory]] is in (\hyperlink{FHT07}{FHT 07, around p. 26}) and \begin{itemize}% \item [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], \emph{Twisted K-theory of differentiable stacks} (\href{http://arxiv.org/abs/math/0306138}{arXiv:math/0306138}) \end{itemize} (which establishes the relation to [[KK-theory]]). \begin{itemize}% \item [[Max Karoubi]], \emph{Twisted bundles and twisted K-theory}, \href{http://arxiv.org/abs/1012.2512}{arxiv/1012.2512} \item [[Ulrich Pennig]], \emph{Twisted K-theory with coefficients in $C^\ast$-algebras}, (\href{http://arxiv.org/abs/1103.4096}{arXiv:1103.4096}) \end{itemize} Discussion in terms of [[vectorial bundles]] is in \begin{itemize}% \item [[Kiyonori Gomi]], \emph{Twisted K-theory and finite-dimensional approximation} (\href{http://arxiv.org/abs/0803.2327}{arXiv}) \item [[Kiyonori Gomi]] und Yuji Terashima, \emph{Chern-Weil Construction for Twisted K-Theory} Communication ins Mathematical Physics, Volume 299, Number 1, 225-254, \end{itemize} The twisted version of [[differential K-theory]] is discussed in \begin{itemize}% \item [[Alan Carey]], \emph{Differential twisted K-theory and applications} ESI preprint (\href{http://www.esi.ac.at/preprints/esi1945.pdf}{pdf}) \end{itemize} Discussion of combined [[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[KR-theory]] on [[orbifold|orbi-]] [[orientifolds]]: \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} Discussion of twisted [[K-homology]]: \begin{itemize}% \item [[Bai-Ling Wang]], \emph{Gemometric cycles, index theory and twisted K-homology} (\href{https://arxiv.org/abs/0710.1625}{arXiv:0710.1625}) \item [[Eckhard Meinrenken]], \emph{Twisted K-homology and group-valued moment maps}, International Mathematics Research Notices 2012 (20) (2012), 4563--4618 (\href{https://arxiv.org/abs/1008.1261}{arXiv:1008.1261}) \item Bei Liu, \emph{Twisted K-homology,Geometric cycles and T-duality} (\href{https://arxiv.org/abs/1411.1575}{arXiv:1411.1575}) \end{itemize} Discussion of combined [[twisted K-theory|twisted]] and [[equivariant K-theory|equivariant]] and [[real K-theory|real]] K-theory \begin{itemize}% \item [[Kiyonori Gomi]], \emph{Freed-Moore K-theory} (\href{https://arxiv.org/abs/1705.09134}{arXiv:1705.09134}) \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} [[!redirects twisted K-theories]] \end{document}