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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*{equivariant 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_operator_ktheory_of_crossed_product_algebras}{Relation to operator K-theory of crossed product algebras}\dotfill \pageref*{relation_to_operator_ktheory_of_crossed_product_algebras} \linebreak \noindent\hyperlink{RelationToRepresentationTheory}{Relation to representation theory}\dotfill \pageref*{RelationToRepresentationTheory} \linebreak \noindent\hyperlink{EquivariantKUAndTheComplexRepresentationRing}{Equivariant $KU$ and the complex representation ring}\dotfill \pageref*{EquivariantKUAndTheComplexRepresentationRing} \linebreak \noindent\hyperlink{ChernClassesOfLinearRepresentations}{Chern classes of linear representations}\dotfill \pageref*{ChernClassesOfLinearRepresentations} \linebreak \noindent\hyperlink{equivariant__and_the_real_representation_ring}{Equivariant $KO$ and the real representation ring}\dotfill \pageref*{equivariant__and_the_real_representation_ring} \linebreak \noindent\hyperlink{relation_to_ktheory_of_homotopy_quotient_spaces_borel_constructions}{Relation to K-theory of homotopy quotient spaces (Borel constructions)}\dotfill \pageref*{relation_to_ktheory_of_homotopy_quotient_spaces_borel_constructions} \linebreak \noindent\hyperlink{equivariant_cherncharacter}{Equivariant Chern-character}\dotfill \pageref*{equivariant_cherncharacter} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{for_dbrane_charge_on_orbifolds}{For D-brane charge on orbifolds}\dotfill \pageref*{for_dbrane_charge_on_orbifolds} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \emph{Equivariant K-theory} is the [[equivariant cohomology]] version of the [[generalized cohomology theory]] [[K-theory]]. To the extent that [[K-theory]] is given by [[equivalence classes]] of [[virtual vector bundles]] ([[topological K-theory]], [[operator K-theory]]), equivariant K-theory is given by equivalence classes of virtual [[equivariant bundles]] or generalizations to [[noncommutative topology]] thereof, as in \emph{[[equivariant operator K-theory]]}, \emph{[[equivariant KK-theory]]}. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_operator_ktheory_of_crossed_product_algebras}{}\subsubsection*{{Relation to operator K-theory of crossed product algebras}}\label{relation_to_operator_ktheory_of_crossed_product_algebras} The \emph{[[Green-Julg theorem]]} identifies, under some conditions, equivariant K-theory with [[operator K-theory]] of corresponding [[crossed product algebras]]. \hypertarget{RelationToRepresentationTheory}{}\subsubsection*{{Relation to representation theory}}\label{RelationToRepresentationTheory} \hypertarget{EquivariantKUAndTheComplexRepresentationRing}{}\paragraph*{{Equivariant $KU$ and the complex representation ring}}\label{EquivariantKUAndTheComplexRepresentationRing} The [[representation ring]] of $G$ over the [[complex numbers]] is the $G$-[[equivariant K-theory]] of the point, or equivalently by the [[Green-Julg theorem]], if $G$ is a [[compact Lie group]], the [[operator K-theory]] of the [[group algebra]] (the [[groupoid convolution algebra]] of the [[delooping]] groupoid of $G$): \begin{equation} R_{\mathbb{C}}(G) \simeq KU^0_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,. \label{RepresentationRingAsEquivariantKTheoryOfThePoint}\end{equation} The first [[isomorphism]] here follows immediately from the elementary definition of equivariant [[topological K-theory]], since a $G$-[[equivariant vector bundle]] over the point is manifestly just a [[linear representation]] of $G$ on a [[complex vector space]]. (e.g. \hyperlink{Greenlees05}{Greenlees 05, section 3}, \hyperlink{Wilson16}{Wilson 16, example 1.6 p. 3}) \hypertarget{ChernClassesOfLinearRepresentations}{}\paragraph*{{Chern classes of linear representations}}\label{ChernClassesOfLinearRepresentations} Under the identification \eqref{RepresentationRingAsEquivariantKTheoryOfThePoint} and the [[Atiyah-Segal completion]] map \begin{displaymath} R_{\mathbb{C}}(G) \simeq KU_G^0(\ast) \overset{ \widehat{(-)} }{\longrightarrow} KU(BG) \end{displaymath} one may ask for the [[Chern character]] of the K-theory class $\widehat{V} \in KU(B G)$ expressed in terms of the actual [[character]] of the [[representation]] $V$. For more see at \emph{[[Chern class of a linear representation]]}. There is a closed formula at least for the [[first Chern class]] (\hyperlink{Atiyah61}{Atiyah 61, appendix}): For 1-dimensional representations $V$ their [[first Chern class]] $c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})$ is their image under the canonical [[isomorphism]] from 1-[[dimension|dimensional]] characters in $Hom_{Grp}(G,U(1))$ to the [[group cohomology]] $H^2_{grp}(G, \mathbb{Z})$ and further to the [[ordinary cohomology]] $H^2(B G, \mathbb{Z})$ of the [[classifying space]] $B G$: \begin{displaymath} c_1\left(\widehat{(-)}\right) \;\colon\; Hom_{Grp}(G, U(1)) \overset{\simeq}{\longrightarrow} H^2_{grp}(G,\mathbb{Z}) \overset{\simeq}{\longrightarrow} H^2(B G, \mathbb{Z}) \,. \end{displaymath} More generally, for $n$-[[dimension|dimensional]] [[linear representations]] $V$ their [[first Chern class]] $c_1(\widehat V)$ is the previously defined first Chern-class of the [[line bundle]] $\widehat{\wedge^n V}$ corresponding to the $n$-th [[exterior power]] $\wedge^n V$ of $V$. The latter is a 1-dimensional representation, corresponding to the [[determinant line bundle]] $det(\widehat{V}) = \widehat{\wedge^n V}$: \begin{displaymath} c_1(\widehat{V}) \;=\; c_1(det(\widehat{V})) \;=\; c_1( \widehat{\wedge^n V} ) \,. \end{displaymath} (\hyperlink{Atiyah61}{Atiyah 61, appendix, item (7)}) More explicitly, via the formula for the [[determinant]] as a [[polynomial]] in [[traces]] of powers (see \href{determinant#eq:DeterminantAsPolynomialInTracesOfPowers}{there}) this means that the first Chern class of the $n$-dimensional representation $V$ is expressed in terms of its [[character]] $\chi_V$ as \begin{equation} c_1(V) = \chi_{\left(\wedge^n V\right)} \;\colon\; g \;\mapsto\; \underset{ { k_1,\cdots, k_n \in \mathbb{N} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } \left(\chi_V(g^l)\right)^{k_l} \label{FirstChernClassOfRepresentationInTermsOfTheCharacter}\end{equation} For example, for a representation of dimension $n = 2$ this reduces to \begin{displaymath} c_1(V) = \chi_{V \wedge V} \;\colon\; g \;\mapsto\; \frac{1}{2} \left( \left( \chi_V(g)\right)^2 - \chi_V(g^2) \right) \end{displaymath} (see also e.g. \href{representation+theory#tomDieck09}{tom Dieck 09, p. 45}) $\,$ \hypertarget{equivariant__and_the_real_representation_ring}{}\paragraph*{{Equivariant $KO$ and the real representation ring}}\label{equivariant__and_the_real_representation_ring} An isomorphism analogous to \eqref{RepresentationRingAsEquivariantKTheoryOfThePoint} identifies the $G$-representation ring over the [[real numbers]] with the equivariant orthogonal $K$-theory of the point in degree 0: \begin{displaymath} R_{\mathbb{R}}(G) \;\simeq\; KO_G^0(\ast) \,. \end{displaymath} But beware that equivariant [[KO]], even of the point, is much richer in higher degree (\hyperlink{Wilson16}{Wilson 16, remark 3.34}). In fact, [[equivariant KO-theory]] of the point subsumes the [[representation rings]] over the [[real numbers]], the [[complex numbers]] and the [[quaternions]]: \begin{displaymath} KO_G^n(\ast) \;\simeq\; \left\{ \itexarray{ 0 &\vert& n = 7 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5 \\ R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) } &\vert& n = 4 \\ 0 &\vert& n = 3 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2 \\ R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1 \\ R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0 } \right. \end{displaymath} (\hyperlink{Greenlees05}{Greenlees 05, p. 3}) Accordingly the construction of an [[index]] ([[push-forward in generalized cohomology|push-forward]] to the point) in equivariant K-theory is a way of producing $G$-[[representations]] from [[equivariant vector bundles]]. This method is also called \emph{[[Dirac induction]]}. Specifically, applied to equivariant [[complex line bundles]] on [[coadjoint orbits]] of $G$, this is a K-theoretic formulation of the [[orbit method]]. \hypertarget{relation_to_ktheory_of_homotopy_quotient_spaces_borel_constructions}{}\subsubsection*{{Relation to K-theory of homotopy quotient spaces (Borel constructions)}}\label{relation_to_ktheory_of_homotopy_quotient_spaces_borel_constructions} For $X$ a [[topological space]] equipped with a $G$-[[action]] for $G$ a [[topological group]], write $X//G$ for the [[homotopy type]] of the corresponding [[homotopy quotient]]. A standard model for this is the [[Borel construction]] \begin{displaymath} X//G \simeq (X \times EG)/G \,. \end{displaymath} The ordinary [[topological K-theory]] of $X//G$ is also called the \emph{Borel-equivariant K-theory} of $X$, denoted \begin{displaymath} K_G^{Bor}(X) \coloneqq K(X//G) \,. \end{displaymath} There is a canonical map \begin{displaymath} K_G(X) \to K_G^{Bor}(X) \end{displaymath} from the genuine equivariant K-theory to the Borel equivariant K-theory. In terms of the [[Borel construction]] this is given by the composite \begin{displaymath} K_G(X) \to K_G(X \times E G) \simeq K((X \times E G) / G ) \simeq K_G^{Bor}(X) \,, \end{displaymath} where the first map is pullback along the [[projection]] $X \times E G \to X$ and the first equivalence holds because the $G$-action on $X \times E G$ is free. This map from genuine to Borel equivariant K-theory is not in general an isomorphism. Specifically for $X$ the point, then $K_G(\ast) \simeq R(G)$ is the [[representation ring]] and $K_G^{Bor}(\ast) \simeq K(B G)$ is the [[topological K-theory]] of the [[classifying space]] $B G$ of $G$-[[principal bundles]]. In this case the above canonical map is of the form \begin{displaymath} R(G) \to K(B G) \,. \end{displaymath} This is never an [[isomorphism]], unless $G$ is the trivial group. But the [[Atiyah-Segal completion theorem]] says that the map identifies $K(B G)$ as the completion of $R(G)$ at the [[ideal]] of [[virtual representations]] of rank 0. [[!include Segal completion -- table]] \hypertarget{equivariant_cherncharacter}{}\subsubsection*{{Equivariant Chern-character}}\label{equivariant_cherncharacter} There is a [[Chern character]] map from equivariant K-theory to [[equivariant ordinary cohomology]]. (e.g. \hyperlink{Stefanich}{Stefanich}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Baum-Connes conjecture]], [[Green-Julg theorem]], \item [[Atiyah-Segal completion theorem]] \item [[equivariant elliptic cohomology]] \item [[equivariant algebraic K-theory]] \item [[McKay correspondence]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The idea of equivariant [[topological K-theory]] and the [[Atiyah-Segal completion theorem]] goes back to \begin{itemize}% \item [[Michael Atiyah]], \emph{Characters and cohomology of finite groups}, Publications Mathématiques de l'IHÉS, Volume 9 (1961) , p. 23-64 (\href{http://www.numdam.org/item?id=PMIHES_1961__9__23_0}{numdam}) \item [[Michael Atiyah]], [[Friedrich Hirzebruch]], \emph{Vector bundle and homogeneous spaces}, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 ([[AtiyahHirzebruch61.pdf:file]]) \item [[Graeme Segal]], \emph{Equivariant K-theory}, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) p. 129-151 \item [[Graeme Segal]], [[Michael Atiyah]], \emph{Equivariant K-theory and completion}, J. Differential Geometry 3 (1969), 1--18. MR 0259946 \end{itemize} and for [[algebraic K-theory]] to \begin{itemize}% \item [[Robert Thomason]], \emph{Algebraic K-theory of group scheme actions}, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539--563 \end{itemize} See also at \emph{\href{algebraic+K-theory#ReferencesAlgebraicKTheoryForQuotientStacks}{algebraic K-theory -- References -- On quotient stacks}}. Introductions and surveys include \begin{itemize}% \item [[John Greenlees]], \emph{Equivariant version of real and complex connective K-theory}, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. (\href{http://projecteuclid.org/euclid.hha/1139839291}{Euclid:1139839291}) \item N. C. Phillips, \emph{Equivariant K-theory for proper actions}, Pitman Research Notes in Mathematics Series 178, Longman, Harlow, UK, 1989. \item [[Bruce Blackadar]], section 11 of \emph{[[K-Theory for Operator Algebras]]} \item Alexander Merkujev, \emph{Equivariant K-theory} (\href{http://www.math.uiuc.edu/K-theory/0981/book/2-0925-0954.pdf}{pdf}) \item Zachary Maddock, \emph{An informal discourse on equivariant K-theory} (\href{http://math.columbia.edu/~ellis/ass/equiv_k-thry.pdf}{pdf}) \item [[Akhil Mathew]], \emph{\href{https://amathew.wordpress.com/2011/12/03/equivariant-k-theory/}{Equivariant K-theory}} \item [[Dylan Wilson]], \emph{Equivariant K-theory}, 2016 (\href{https://www.math.uchicago.edu/~dwilson/notes/equivariant-k-theory-talk.pdf}{pdf}, [[WilsonKTheory16.pdf:file]]) \end{itemize} The equivariant [[Chern character]] is discussed in \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} Discussion relating to K-theory of [[homotopy quotients]]/[[Borel constructions]] is in \begin{itemize}% \item [[Jacob Lurie]], around p. 11 of \emph{[[A Survey of Elliptic Cohomology]]} (\href{http://www.math.harvard.edu/~lurie/papers/survey.pdf}{pdf}) \end{itemize} Discussion of the [[adjoint action]]-equivariant K-theory of suitable [[Lie groups]] in in \begin{itemize}% \item [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], \emph{[[Loop Groups and Twisted K-Theory]]}. \end{itemize} Discussion of K-theory of [[orbifolds]] is for instance in section 3 of \begin{itemize}% \item Alejandro Adem, Johanna Leida, Yongbin Ruan, \emph{Orbifolds and string topology}, Cambridge Tracts in Mathematics 171, 2007 (\href{http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf}{pdf}) \end{itemize} Discussion of combined [[twisted K-theory|twisted]] and [[equivariant K-theory|equivariant]] and [[real K-theory|real]] K-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} \hypertarget{for_dbrane_charge_on_orbifolds}{}\subsubsection*{{For D-brane charge on orbifolds}}\label{for_dbrane_charge_on_orbifolds} The proposal that [[D-brane charge]] on [[orbifolds]] is given by [[equivariant K-theory]] goes back to \begin{itemize}% \item [[Edward Witten]], section 5.1 of \emph{D-Branes And K-Theory}, JHEP 9812:019,1998 (\href{http://arxiv.org/abs/hep-th/9810188}{arXiv:hep-th/9810188}) \end{itemize} but it was pointed out that only a subgroup or quotient group of equivariant K-theory can be physically relevant, in \begin{itemize}% \item [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], around (137) of \emph{Triples, Fluxes, and Strings}, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (\href{https://arxiv.org/abs/hep-th/0103170}{arXiv:hep-th/0103170}) \end{itemize} For further references see at \emph{[[fractional D-brane]]}. On [[Chern classes of linear representations]]: \begin{itemize}% \item \hyperlink{Atiyah61}{Atiyah 61, Appendix} \item Leonard Evens, \emph{On the Chern Classes of Representations of Finite Groups}, Transactions of the American Mathematical Society Vol. 115 (Mar., 1965), pp. 180-193 (\href{https://www.jstor.org/stable/1994264}{doi:10.2307/1994264}) \item F. Kamber, Ph. Tondeur, \emph{Flat Bundles and Characteristic Classes of Group-Representations}, American Journal of Mathematics Vol. 89, No. 4 (Oct., 1967), pp. 857-886 (\href{https://www.jstor.org/stable/2373408}{doi:10.2307/2373408}) \item [[Peter Symonds]], \emph{A splitting principle for group representations}, Comment. Math. Helv. (1991) 66: 169 (\href{https://doi.org/10.1007/BF02566643}{doi:10.1007/BF02566643}) \end{itemize} [[!redirects equivariant KO-theory]] \end{document}