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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*{fiber integration in K-theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory} [[!include index theory - contents]] \hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory} [[!include integration theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{Models}{Models}\dotfill \pageref*{Models} \linebreak \noindent\hyperlink{InTermsOfBundlesOfFredholmOperators}{In terms of bundles of Fredholm operators}\dotfill \pageref*{InTermsOfBundlesOfFredholmOperators} \linebreak \noindent\hyperlink{along_a_fibration_of_closed_manifolds}{Along a fibration of closed $Spin^c$-manifolds}\dotfill \pageref*{along_a_fibration_of_closed_manifolds} \linebreak \noindent\hyperlink{InOperatorKKTheory}{In operator KK-theory}\dotfill \pageref*{InOperatorKKTheory} \linebreak \noindent\hyperlink{KKPushForwardAlongEmbedding}{Along an embedding}\dotfill \pageref*{KKPushForwardAlongEmbedding} \linebreak \noindent\hyperlink{KKPushforwardAlongSubmersion}{Along a proper submersion}\dotfill \pageref*{KKPushforwardAlongSubmersion} \linebreak \noindent\hyperlink{AlongAFibrationOfClosedSpinCManifolds}{Along a smooth fibration of closed $Spin^c$-manifolds}\dotfill \pageref*{AlongAFibrationOfClosedSpinCManifolds} \linebreak \noindent\hyperlink{#KKPushforwardAlongGeneralMap}{Along a general K-oriented map}\dotfill \pageref*{#KKPushforwardAlongGeneralMap} \linebreak \noindent\hyperlink{KKPushforwardAlongGeneralMap}{In twisted K-theory}\dotfill \pageref*{KKPushforwardAlongGeneralMap} \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} The special case of [[fiber integration in generalized cohomology]]/[[twisted Umkehr maps]] for [[KU]]-cohomology. \hypertarget{Models}{}\subsection*{{Models}}\label{Models} There are various different models for describing and constructing fiber integration in K-theory. \begin{enumerate}% \item \hyperlink{InTermsOfBundlesOfFredholmOperators}{In terms of bundles of Fredholm operators} \item \hyperlink{InOperatorKKTheory}{In operator KK-theory} \end{enumerate} \hypertarget{InTermsOfBundlesOfFredholmOperators}{}\subsubsection*{{In terms of bundles of Fredholm operators}}\label{InTermsOfBundlesOfFredholmOperators} We discuss here fiber integration in the model of [[twisted K-theory]] \href{twisted%20K-theory#ByBundlesOfFredholmOperators}{by bundles of spaces of Fredholm operators}. Related literature includes (\href{CareyWang05}{Carey-Wang 05}). \begin{enumerate}% \item \emph{\hyperlink{AlongAFibrationOfClosedSpinCManifolds}{Along a fibration of closed spin{\tt \symbol{94}}c manifolds}} \end{enumerate} \hypertarget{along_a_fibration_of_closed_manifolds}{}\paragraph*{{Along a fibration of closed $Spin^c$-manifolds}}\label{along_a_fibration_of_closed_manifolds} Let $f\colon Y \longrightarrow X$ be a [[fiber bundle]] of [[compact manifolds|compact]] [[smooth manifolds]] carrying fiberwise a [[spin{\tt \symbol{94}}c structure]]. For $x\in X$ write \begin{itemize}% \item $Y_x$ for the [[fiber]] over $x$; \item $P_x \to Y_x$ for the fiberwise [[spin{\tt \symbol{94}}c]]-[[principal bundle]]; \item $Cl(Y)_x \coloneqq P_x \underset{Spin^c}{\times} Cl_n$ for the fiberwise [[Clifford bundle]]; \item $S_x$ for the [[spin{\tt \symbol{94}}c]] [[spin bundle]] on $Y_x$; \item $D_x\colon \Gamma(S_x)\to \Gamma(S_x)$ for the [[Spin{\tt \symbol{94}}c Dirac operator]] on the [[space of sections]] of [[spinors]]; \item $\tilde D_x \coloneqq \frac{D_x}{\sqrt{1+ D_x^\ast D_x}} \colon L^2(S_x)\to L^2(S_x)$ for the [[bounded linear operator|bounded]] version; \end{itemize} Here $D_x$ depends smoothly on $x$ while $\tilde D_x$ still depends continuously on $x$. Equip $CL(Y)_y$ with the $Cl_n$-[[action]] given on elements $v \in \mathbb{R}^n \hookrightarrow Cl_n$ joint right Clifford product by $v$ and left Clifford product by the volume element \begin{displaymath} vol \cdot (-) \cdot v \,. \end{displaymath} This is such that $D_x$ [[graded commutator|graded-commutes]] with this $Cl_n$-action. Hence the assignment $x \mapsto \tilde D_x$ defines a map \begin{displaymath} \tilde D_{(-)} \colon X \longrightarrow Fred^{(n)} \end{displaymath} from the base to the space [[Fredholm operators]] [[graded-commutator|graded-commuting]] with $Cl_n$, as defined \href{twisted%20K-theory#ByBundlesOfFredholmOperators}{here} at \emph{[[twisted K-theory]]}. More generally, let then $V \to Y$ be a [[vector bundle]] representing a class in $K^0(Y)$. With a choice of [[connection]] this twists the above constrction to yield $V$-twisted Dirac operator $\tilde D^V_x$ and hence a map \begin{displaymath} \tilde D_{(-)}^V \colon X \longrightarrow Fred^{(n)} \,. \end{displaymath} This represents the push-forward class in $K^{dim(Y_x)}(X)$, and this construction gives a map \begin{displaymath} \int f \colon K^0(Y) \longrightarrow K^{dim(Y)-dim()X}(X) \,. \end{displaymath} For $dim(Y_x)$ even and hence ignoring the compatibility with the $Cl_n$-action, this is discussed in (\href{CareyWang05}{Carey-Wang 05}). \hypertarget{InOperatorKKTheory}{}\subsubsection*{{In operator KK-theory}}\label{InOperatorKKTheory} We discuss fiber integration /push-forward/[[Umkehr maps]]/[[Gysin maps]] in [[operator K-theory]], hence in [[KK-theory]] (\hyperlink{ConnesSkandalis84}{Connes-Skandalis 85}, \hyperlink{BMRS07}{BMRS 07, section 3}). The following discusses KK-pushforward \begin{enumerate}% \item \emph{\hyperlink{KKPushForwardAlongEmbedding}{Along an embedding}} \item \emph{\hyperlink{KKPushforwardAlongSubmersion}{Along a submersion}} \item \emph{\hyperlink{AlongAFibrationOfClosedSpinCManifolds}{Along a fibration of closed spin{\tt \symbol{94}}c manifolds}} \item \emph{\hyperlink{KKPushforwardAlongGeneralMap}{Along a general K-oriented map}} \item \emph{\hyperlink{KKPushforwardAlongGeneralMap}{In twisted K-theory}} \end{enumerate} The construction goes back to (\hyperlink{Connes82}{Connes 82}), where it is given over smooth manifolds. Then (\hyperlink{ConnesSkandalis84}{Connes-Skandalis 84}, \hyperlink{HilsumSkandalis87}{Hilsum-Skandalis 87}) generalize this to maps between [[foliations]] by KK-elements betwen the [[groupoid convolution algebras]] of the coresponding [[holonomy groupoids]] and (\hyperlink{RouseWang10}{Rouse-Wang 10}) further generalize to the case where a [[circle 2-bundle]] twist is present over these foliations. A purely algebraic generalization to (K-oriented) maps between otherwise arbitrary [[noncommutative topology|noncommutative spaces]]/[[C\emph{-algebras]] is in (\hyperlink{BMRS07}{BMRS 07}).} \hypertarget{KKPushForwardAlongEmbedding}{}\paragraph*{{Along an embedding}}\label{KKPushForwardAlongEmbedding} (\href{ConnesSkandalis84}{Connes-Skandalis 84, above prop. 2.8}) Let $h \colon X \hookrightarrow Y$ be an [[embedding]] of [[compact topological space|compact]] [[smooth manifolds]]. The push-forward constructed from this is supposed to be an element in [[KK-theory]] \begin{displaymath} h! \colon KK_d(C(X), C(Y)) \end{displaymath} in terms of which the push-forward on [[operator K-theory]] is induced by postcomposition: \begin{displaymath} h_! \;\colon\; K^\bullet(X) \simeq KK_\bullet(\mathbb{C}, X) \stackrel{h!\circ (-)}{\to} KK_{\bullet+d}(\mathbb{C},Y) \simeq KK^{\bullet+d}(Y) \,, \end{displaymath} where $d = dim(X) - dim(Y)$. Now, if we could ``thicken'' $X$ a bit, namely to a [[tubular neighbourhood]] \begin{displaymath} h \;\colon\; X \hookrightarrow U \stackrel{j}{\hookrightarrow} Y \end{displaymath} of $h(X)$ in $Y$ without changing the K-theory of $X$, then the element in question will just be the KK-element \begin{displaymath} j! \in KK(C_0(U), C(Y)) \end{displaymath} induced directly from the [[C\emph{-algebra]] homomorphism $C_0(U) \to C(Y)$ from the [[algebra of functions]] [[vanishing at infinity]] of $U$ to functions on $Y$, given by extending these functions by 0 to functions on $Y$. Or rather, it will be that element composed with the assumed KK-equivalence} \begin{displaymath} \psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,. \end{displaymath} The bulk of the technical work in constructing the push-forward is in constructing this equivalence. (\hyperlink{BMRS07}{BMRS 07, example 3.3}) In order for it to exist at all, assume that the [[normal bundle]] \begin{displaymath} N_Y X \coloneqq h^\ast(T Y)/ T X \end{displaymath} has a [[spin{\tt \symbol{94}}c structure]]. Write $S(N_Y X)$ for the associated [[spinor bundle]]. Then there is an invertible element in [[KK-theory]] \begin{displaymath} \iota^X! \in KK_n(C(X), C_0(N_Y X)) \end{displaymath} hence a KK-equivalence $\iota^X! \colon C(X) \stackrel{\simeq}{\to} C_0(N_Y X)$, where $C_0(-)$ denotes the [[algebra of functions]] [[vanishing at infinity]]. This is defined as follows. Consider the pullback $\pi_n^\ast S(N_Y X) \to N_Y X$ of this spinor to the normal bundle itself along the projection $\pi_N \colon N_Y X \to X$. Then\ldots{} Moreover, a choice of a [[Riemannian metric]] on $X$ allows to find a [[diffeomorphism]] between the [[tubular neighbourhood]] $U_{h(X)}$ of $h(X)$ and a neighbourhood of the zero-section of of the normal bundle \begin{displaymath} \Phi \colon U_{h(X)} \hookrightarrow N_Y X \,. \end{displaymath} This induces a KK-equivalence \begin{displaymath} [\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,. \end{displaymath} Therefore the push-forward in operator K-theory along $f \colon X \hookrightarrow Y$ is given by postcomposing in [[KK-theory]] with \begin{displaymath} h! \colon C(X) \underoverset{\simeq_{KK}}{i^X!}{\to} C_0(N_Y X) \underoverset{\simeq_{KK}}{\Phi}{\to} C_0(U) \stackrel{j!}{\to} C(Y) \,. \end{displaymath} \hypertarget{KKPushforwardAlongSubmersion}{}\paragraph*{{Along a proper submersion}}\label{KKPushforwardAlongSubmersion} (\href{ConnesSkandalis84}{Connes-Skandalis 84, above prop. 2.9}) For $\pi \colon X \to Z$ a [[K-orientation|K-oriented]] [[proper map|proper]] [[submersion]] of compact smooth manifolds, the push-forward map along it is reduced to the \hyperlink{KKPushForwardAlongEmbedding}{above} case of an embedding by \begin{enumerate}% \item using that by the [[Whitney embedding theorem]] every compact $X$ may be embedded into some $\mathbb{R}^{2q}$ such as to yield an embedding \begin{displaymath} h \colon X \to Z \times \mathbb{R}^{2 q} \end{displaymath} \item using that there is a KK-equivalence \begin{displaymath} \iota^Z! \colon C(Z) \stackrel{\simeq_{KK}}{\to} C_0(Z \times \mathbb{R}^{2q}) \,. \end{displaymath} \end{enumerate} The resulting push-forward is then given by postcomposition in [[KK-theory]] with \begin{displaymath} \pi! \colon C(X) \stackrel{h!}{\to} C_0(Z \times \mathbb{R}^{2}q) \underoverset{\simeq_{KK}}{(\iota^Z!)^{-1}}{\to} C(Z) \,. \end{displaymath} (\hyperlink{BMRS07}{BMRS 07, example 3.4}) \hypertarget{AlongAFibrationOfClosedSpinCManifolds}{}\paragraph*{{Along a smooth fibration of closed $Spin^c$-manifolds}}\label{AlongAFibrationOfClosedSpinCManifolds} Specifically, for $\pi \colon X \to Z$ a smooth fibration over a closed smooth manifold whose [[fibers]] $X/Z$ are \begin{itemize}% \item [[closed manifold|closed]] [[smooth manifolds|smooth]] [[spin{\tt \symbol{94}}c structure|spin{\tt \symbol{94}}c]] [[manifolds]] of even [[dimension]] \end{itemize} the push-forward element $\pi! \in KK(C_0(X), C_0(Z))$ is given by the [[Fredholm module|Fredholm]]-[[Hilbert module]] obatined from the fiberwise [[spin{\tt \symbol{94}}c Dirac operator]] acting on the fiberwise [[spinors]]. (\href{ConnesSkandalis84}{Connes-Skandalis 84, proof of lemma 4.7}, \href{BMRS07}{BMRS 07, example 3.9}). In detail, write \begin{displaymath} T(X/Z) \hookrightarrow T X \end{displaymath} for the sub-bundle of the total [[tangent bundle]] on the [[vertical vectors]] and choose a [[Riemannian metric]] $g^{X/Z}$ on this bundle (hence a collection of Riemannian metric on the fibers $X/Z$ smoothly varying along $Z$). Write $S_{X/Z}$ for the corresponding [[spinor bundle]]. A choice of horizontal complenet $T X \simeq T^H X \oplus T(X/Z)$ induces an [[affine connection]] $\nabla^{X/Z}$. This combined with the [[symbol map]]/Clifford multiplication of $T^\ast (X/Z)$ on $S_{X/Z}$ induces a fiberwise [[spin{\tt \symbol{94}}c Dirac operator]], acting in each fiber on the [[Hilbert space]] $L^2(X/Z, S_{X/Z})$. This yields a [[Fredholm module|Fredholm]]-[[Hilbert bimodule]] \begin{displaymath} (D_{X/Z}, L^2(X/Z, S_{X/Z})) \end{displaymath} which defines an element in [[KK-theory]] \begin{displaymath} \pi ! \in KK(C_0(X), C_0(Z)) \,. \end{displaymath} Postcompositon with this is the push-forward map in K/KK-theory, equivalently the [[index]] map of the collection of Dirac operators. \hypertarget{KKPushforwardAlongGeneralMap}{}\paragraph*{{Along a general K-oriented map}}\label{KKPushforwardAlongGeneralMap} (\href{ConnesSkandalis84}{Connes-Skandalis 84, def. 2.1}) Now for $f \colon X \to Y$ an arbitray [[K-orientation|K-oriented]] smooth proper map, we may reduce push-forward along it to the above two cases by factoring it through its [[graph map]], followed by projection to $Y$: \begin{displaymath} f \;\colon\; X \stackrel{graph(f)}{\to} X \times Y \stackrel{p_Y}{\to} Y \,. \end{displaymath} Hence push-forward along such a general map is postcomposition in [[KK-theory]] with \begin{displaymath} f! \coloneqq p_Y !\circ graph(f)! \,. \end{displaymath} (\hyperlink{BMRS07}{BMRS 07, example 3.5}) \hypertarget{KKPushforwardAlongGeneralMap}{}\paragraph*{{In twisted K-theory}}\label{KKPushforwardAlongGeneralMap} We discuss push forward in K-theory more generally by [[PoincarĂ© duality C\emph{-algebras]] hence [[dual objects]] in [[KK-theory]].} Let $i \colon Q \to X$ be a map of [[compact topological space|compact]] [[manifolds]] and let $\chi \colon X \to B^2 U(1)$ modulate a [[circle 2-bundle]] regarded as a [[twisted K-theory|twist for K-theory]]. Then forming [[twisted groupoid convolution algebras]] yields a [[KK-theory]] morphism of the form \begin{displaymath} C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\longleftarrow} C_{\chi}(X) \,, \end{displaymath} with notation as in \href{Poincaré+duality+algebra#CStarAlgebraOf2BundleOnManifold}{this definition}. By \href{Poincaré+duality+algebra#DualOfCompactManifoldWithTwist}{this proposition} the [[dual morphism]] is of the form \begin{displaymath} C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{W_3(\tau_X)}{\chi}}(X) \,. \end{displaymath} If we assume that $X$ has a [[spin{\tt \symbol{94}}c structure]] then this is \begin{displaymath} C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{1}{\chi}}(X) \,. \end{displaymath} Postcomposition with this map in [[KK-theory]] now yields a map from the $\frac{W_3(\tau_Q)}{i^\ast \chi}$-[[twisted K-theory]] of $Q$ to the $\chi^{-1}$-[[twisted K-theory]] of $X$: \begin{displaymath} i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi} \,. \end{displaymath} If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a [[D-brane]] [[worldvolume]], then $\chi$ would be the class of the [[background gauge field|background]] [[B-field]] and an element \begin{displaymath} [\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \end{displaymath} is called (the K-class of) a \emph{[[Chan-Paton gauge field]]} on the D-brane satisfying the \emph{[[Freed-Witten-Kapustin anomaly cancellation]]} mechanism. (The orginal \emph{[[Freed-Witten anomaly cancellation]]} assumes $\xi$ given by a [[twisted unitary bundle|twisted line bundle]] in which case it exhibits a [[twisted spin{\tt \symbol{94}}c structure]] on $Q$.) Finally its [[fiber integration|push-forward]] \begin{displaymath} [i_! \xi] \in K_{\bullet- \chi}(X) \end{displaymath} is called the corresponding \emph{[[D-brane charge]]}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[geometric quantization by push-forward]] \item [[geometric quantization with KU-coefficients]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Discussion in [[KK-theory]] \begin{itemize}% \item [[Alain Connes]], \emph{A survey of foliations and operator algebras}, Proceedings of the A.M.S., 38, 521-628 (1982) (\href{http://www.alainconnes.org/docs/foliationsfine.pdf}{pdf}) \item [[Alain Connes]], [[Georges Skandalis]], \emph{The longitudinal index theorem for foliations}. Publ. Res. Inst. Math. Sci. 20, no. 6, 1139--1183 (1984) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.4218}{web}) \item [[Michel Hilsum]], [[Georges Skandalis]], \emph{Morphismes K-orient\'e{} d'espace de feuille et fonctoralit\'e{} en th\'e{}orie de Kasparov}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 20 no. 3 (1987), p. 325-390 (\href{http://www.numdam.org/item?id=ASENS_1987_4_20_3_325_0}{numdam}) \item Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{Noncommutative correspondences, duality and D-branes in bivariant K-theory}, Adv. Theor. Math. Phys.13:497-552,2009 (\href{http://arxiv.org/abs/0708.2648}{arXiv:0708.2648}) \item Paulo Carrillo Rouse, [[Bai-Ling Wang]], \emph{Twisted longitudinal index theorem for foliations and wrong way functoriality} (\href{http://arxiv.org/abs/1005.3842}{arXiv:1005.3842}) \end{itemize} Discussion for integration [[twisted K-theory]] over [[manifolds]]: \begin{itemize}% \item [[Alan Carey]], [[Bai-Ling Wang]], \emph{Thom isomorphism and Push-forward map in twisted K-theory} (\href{http://arxiv.org/abs/math/0507414}{arXiv:0507414}) \end{itemize} Discussion for integration of [[twisted K-theory]] along [[representable morphisms]] of [[local quotient stacks]]: \begin{itemize}% \item [[Daniel Freed]], [[Mike Hopkins]], [[Constantin Teleman]], \emph{[[Loop Groups and Twisted K-Theory]]} I (2011) (\href{http://arxiv.org/abs/0711.1906}{arXiv:0711.1906}), \end{itemize} Review in the context of [[geometric quantization with KU-coefficients]] is in \begin{itemize}% \item [[Joost Nuiten]], section 4 of \emph{[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]}, 2013 \end{itemize} [[!redirects fiber integration in KU-theory]] [[!redirects push-forward in K-theory]] \end{document}