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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*{index} [[!redirects Index theory]] [[!redirects index theory]] \begin{quote}% This page is about the notion of index in [[analysis]]/[[operator algebra]]. For other notions see elsewhere. \end{quote} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory} [[!include index theory - contents]] \hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis} [[!include functional analysis - contents]] \hypertarget{operator_algebra}{}\paragraph*{{Operator algebra}}\label{operator_algebra} [[!include AQFT and operator algebra contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{IdeaForDifferentialOperators}{For elliptic differential and Fredholm operators}\dotfill \pageref*{IdeaForDifferentialOperators} \linebreak \noindent\hyperlink{ForDiracOperators}{For Dirac operators}\dotfill \pageref*{ForDiracOperators} \linebreak \noindent\hyperlink{GeneralIdeaInKKTheory}{General abstract definition in KK-theory}\dotfill \pageref*{GeneralIdeaInKKTheory} \linebreak \noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} The notion of \emph{index} was originally defined \begin{itemize}% \item \emph{\hyperlink{IdeaForDifferentialOperators}{For elliptic differential operators}} \end{itemize} as an invariant correction of the [[kernel]] of such an operator (namely corrected by the [[cokernel]]). The definition has particularly nice properties in the special case \begin{itemize}% \item \emph{\hyperlink{ForDiracOperators}{For Dirac operators}} \end{itemize} where it coincides with the [[partition function]] of [[supersymmetric quantum mechanics]]. More generally the resulting notion is abstractly characterized as being the pairing operation ([[composition]]) \begin{itemize}% \item \emph{\hyperlink{GeneralIdeaInKKTheory}{in KK-theory}}. \end{itemize} Even more generally, in [[generalized cohomology theory]] indices are given by [[genera]] and universal [[orientation in generalized cohomology]], such as for instance the [[elliptic genus]] for [[elliptic cohomology]] and the [[Witten genus]] for [[tmf]]. See at \emph{[[genus]]} for more on this generalized notion of indices. There is also a quite general [[microlocal formulation of index theory]] due to Kashiwara and Schapira. \hypertarget{IdeaForDifferentialOperators}{}\subsubsection*{{For elliptic differential and Fredholm operators}}\label{IdeaForDifferentialOperators} The \emph{[[analytical index]]} of an [[elliptic operator|elliptic]] [[differential operator]] $D \colon \Gamma(E_1) \to \Gamma(E_2)$ is defined to be the the difference between the [[dimension]] of its [[kernel]] and that of its [[cokernel]]. One reason why this is an interesting invariant of an elliptic differential operator is that when deforming the operator by a [[compact operator]] then the dimension of both the kernel and the cokernel may change, but their difference remains the same. Hence one may think of the analytic index as a ``corrected'' version of its [[kernel]], such as to make it be more invariant. On the other hand, the \emph{[[topological index]]} of an elliptic differential operator $D$, is defined to be the pairing of the [[cup product]] of its [[Chern character]] and the [[Todd class]] of the base manifold with its [[fundamental class]]. More generally such analytic and topological indices are defined for [[Fredholm operators]]. The \emph{[[Atiyah-Singer index theorem]]} assert that these two notios of index are in fact equal. \hypertarget{ForDiracOperators}{}\subsubsection*{{For Dirac operators}}\label{ForDiracOperators} If the [[Fredholm operator]] in question happens to be a \emph{[[Dirac operator]]} $D$ (such as that encoding the [[dynamics]] of a [[spinning particle]] or more generally the [[supercharge]] of a system in [[supersymmetric quantum mechanics]]) then the index of $D$ coincides with the [[partition function]] of this quantum mechanical system, namely the [[super-trace]] of the [[heat kernel]] $\exp(-t D^2)$ of the corresponding [[Hamiltonian]] [[Laplace operator]] $D^2$ (\hyperlink{BerlineGetzlerVergne04}{Berline-Getzler-Vergne 04}). \begin{prop} \label{}\hypertarget{}{} Let $(X,g)$ be a [[compact topological space|compact]] [[Riemannian manifold]] and $\mathcal{E}$ a smooth [[super vector bundle]] and indeed a [[Clifford module bundle]] over $X$. Consider a [[Dirac operator]] \begin{displaymath} D \colon \Gamma(X,\mathcal{E}) \to \Gamma(X, \mathcal{E}) \end{displaymath} with components (with respect to the $\mathbb{Z}_2$-[[graded vector space|grading]]) to be denoted \begin{displaymath} D = \left[ \itexarray{ 0 & D^- \\ D^+ & 0 } \right] \,, \end{displaymath} where $D^- = (D^+)^\ast$. Then $D^+$ is a [[Fredholm operator]] and its index is the [[supertrace]] of the [[kernel]] of $D$, as well as of the [[heat kernel]] of $D^2$: \begin{displaymath} \begin{aligned} ind(D^+) & \coloneqq dim(ker(D^+)) - dim(coker(D^+)) \\ & = dim(ker(D^+)) - dim(ker(D^-)) \\ & = sTr(ker(D)) \\ & = sTr( \exp(-t \, D^2) ) \;\;\; \forall t \gt 0 \end{aligned} \,. \end{displaymath} \end{prop} This appears as (\href{BerlineGetzlerVergne04}{Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50}), based on (\hyperlink{MacKeanSinger67}{MacKean-Singer 67}). \begin{proof} The last step here follows from an argument which is as simple as it is paramount whenever anything involves [[supersymmetry]]: the point is that if a (hermitean) operator $H$ has a [[supercharge]] $D$, in that $H = D^2$, then all its non-vanishing [[eigenstates]] appear in ``[[supermultiplet]]'' pairs of the same eigenvalue: if $|\psi\rangle$ has [[eigenvalue]] $E \gt 0$ under $H$, then \begin{enumerate}% \item $D |\psi\rangle \neq 0$ (since $D D |\psi\rangle = H |\psi\rangle = E |\psi\rangle \neq 0$); \item also $D |\psi\rangle$ has eignevalue $E$ (since $[H,D] = 0$). \end{enumerate} Therefore all eigenstates for non-vanishing eigenvalues appear in pairs whose members have opposite sign under the supertrace. So only states with $H |\psi\rangle = 0$ contribute to the supertrace. But if $H$ and $D$ are hermitean operators for a non-degenerate inner product, then it follows that $(D^2 |\psi\rangle = 0) \Leftrightarrow (D|\psi\rangle = 0)$ and hence these are precisely the states which are also annihilated by the supercharge (are in the kernel of $D$), hence are precisely only the \emph{supersymmetric states}. On these now the weight $\exp(- t D^2) = 1$ and hence the supertrace over this ``Euclidean propagator'' simply counts the number of supersymmetric states, signed by their fermion number. \end{proof} \begin{remark} \label{}\hypertarget{}{} If one thinks of $D^2$ as the time-evolution [[Hamiltonian]] of a system of [[supersymmetric quantum mechanics]] with $D$ the [[supercharge]] on the [[worldline]], then $ker(D)$ is the space of supersymmetric [[quantum states]], $\exp(-t \, D^2)$ is the Euclidean time evolution operator and its [[supertrace]] is the [[partition function]] of the system. Hence we have the translation \begin{itemize}% \item index = partition function . \end{itemize} This kind of argument appears throughout supersymmetric quantum field theory. In dimension 2 it controls the nature of the [[Witten genus]]. \end{remark} \hypertarget{GeneralIdeaInKKTheory}{}\subsubsection*{{General abstract definition in KK-theory}}\label{GeneralIdeaInKKTheory} The abstract [[universal property|universal]] characterization of indices is: the index is the \emph{pairing} in [[KK-theory]]/[[E-theory]]. More in detail, by the discussion there [[KK-theory]] ([[E-theory]]) is the [[category]] $KK$ which is the additive and split exact [[localization]] of the category [[C\emph{Alg]] of [[C}-algebras]] at the [[compact operators]]. For $\mathbb{C}$ the base [[C\emph{-algebra]] of [[complex numbers]] the [[morphisms]] in this category have the following equivalent meaning:} \begin{itemize}% \item [[morphisms]] $\mathbb{C} \to A$ are [[operator K-theory|operator K-cohomology]] classes which are represented by ``[[vector bundles]] over the space represented by $A$'', namely by [[Hilbert modules]] $E$ over $A$; \item [[morphisms]] $A \to \mathcal{C}$ are [[K-homology]] classes which are represented by [[Fredholm operators]] $D$; \item the [[composition]] \begin{displaymath} ind(D_E) \;\colon\; \mathbb{C} \stackrel{E}{\to} A \stackrel{D}{\to} \mathbb{C} \;\;\;\; \in KK(\mathbb{C}, \mathbb{C}) \simeq \mathbb{Z} \end{displaymath} in the category $KK$ (hence the Kasparov product) is the \emph{index} of the Fredholm operator $D$ twisted by $E$. \end{itemize} More generally, if $B$ is some other chosen base [[C\emph{-algebra]] then $KK(A,B)$ is the group of [[Fredholm operators]] $D$ on [[Hilbert module]] bundles over the [[C}-algebra]] $B$, and one takes the pairing \begin{displaymath} ind \coloneqq \circ_{\mathbb{C}, A, B} \;\colon\; KK(\mathbb{C},A) \times KK(A,B) \to KK(\mathbb{C}, B) \end{displaymath} to be the index map relative $B$. (See e.g. \hyperlink{Schick05}{Schick 05, section 6}.) This is the case that the [[Mishchenko-Fomenko index theorem]] applies to. And hence even more generally one may regard \emph{any} composition in $KK$ as as a generalized index map. Via the universal characterizatin of $KK$ itself, this then gives a fundamental and general abstract characterization of the notion of index: \emph{The index pairing is the [[composition]] operation in the [[KK-theory|KK-]][[localization of a category|localization]] of [[C\emph{Alg]], hence in [[noncommutative stable homotopy theory]].}} \hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages} \begin{itemize}% \item [[topological index]] \item [[analytical index]] \item [[fiber integration in K-theory]] \item [[index of a Dirac operator]] \item [[index theorem]] \begin{itemize}% \item [[Atiyah-Singer index theorem]] \item [[Mishchenko-Fomenko index theorem]] \item [[Baum-Connes conjecture]] \end{itemize} \item [[zeta function of an elliptic differential operator]], [[eta function]] \item [[Poincaré–Hopf index theorem]] \item [[Gauss-Bonnet theorem]] \end{itemize} [[!include genera and partition functions - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} Original articles include \begin{itemize}% \item [[Michael Atiyah]], [[Isadore Singer]], \emph{Index theory for skew-adjoint Fredholm operators} (\href{http://www.maths.ed.ac.uk/~aar/papers/askew.pdf}{pdf}) \end{itemize} (\ldots{}) Lecture notes include \begin{itemize}% \item [[Jean-Michel Bismut]], \emph{Introduction to index theory}, lecture notes (\href{http://www.math.leidenuniv.nl/~ajavanp/Notes_Index_TheoryBotao.pdf}{pdf}) \end{itemize} A general introduction with an emphasis of indices as [[Gysin maps]]/[[fiber integration]]/[[Umkehr maps]] is in \begin{itemize}% \item [[Chris Kottke]], \emph{Talbot Workshop 2010 Talk 2: K-Theory and Index Theory} (\href{http://arxiv.org/abs/1010.5002}{arXiv:1010.5002}) \end{itemize} Textbook accounts include chapter III of \begin{itemize}% \item [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], \emph{[[Spin geometry]]}, Princeton University Press (1989) \end{itemize} A standard textbook account of the description of indices of [[Dirac operators]] as [[partition functions]] in [[supersymmetric quantum mechanics]] is \begin{itemize}% \item [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], \emph{Heat Kernels and Dirac Operators}, Springer Verlag Berlin (2004) \end{itemize} based on original articles including \begin{itemize}% \item H. MacKean, [[Isadore Singer]], \emph{Curvature and eigenvalues of the Laplacian}, J. Diff. Geom. 1 (1967) \end{itemize} \begin{itemize}% \item [[Michael Atiyah]], [[Raoul Bott]], V. K. Patodi, \emph{On the heat equation and the index theorem}, Invent. Math. 19 (1973), 279--330 \end{itemize} \begin{itemize}% \item [[Luis Alvarez-Gaumé]], \emph{Supersymmetry and the Atiyah-Singer index theorem}, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (\href{http://projecteuclid.org/euclid.cmp/1103940278}{Euclid}) \end{itemize} \begin{itemize}% \item [[Ezra Getzler]], \emph{Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem}, Comm. Math. Phys. 92 (1983), 163-178. (\href{http://math.northwestern.edu/~getzler/Papers/1103940796.pdf}{pdf}) \end{itemize} \begin{itemize}% \item [[D. Quillen]], \emph{Superconnections and the Chern character} Topology 24 (1985), no. 1, 89--95; \item [[Varghese Mathai]], [[Daniel Quillen]], \emph{Superconnections, Thom classes, and equivariant differential forms}. Topology 25 (1986), no. 1, 85--110; \item [[Ezra Getzler]], \emph{A short proof of the Atiyah-Singer index theorem}, Topology 25 (1986), 111-117 (\href{http://math.northwestern.edu/~getzler/Papers/local.pdf}{pdf}) \end{itemize} \begin{itemize}% \item [[D. Quillen]], \emph{Superconnection character forms and the Cayley transform}. Topology 27 (1988), no. 2, 211--238 \end{itemize} For the more general discussion of indices of [[elliptic complexes]] see \begin{itemize}% \item [[Peter Gilkey]], \emph{Invariance theory, the heat equation, and the Atiyah-Singer index theorem} (\href{http://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf}{pdf}) \end{itemize} An explicit formula in [[Chern-Weil theory]] for indices of differential operators on [[Hilbert modules]]-bundles is discussed in detail in \begin{itemize}% \item [[Thomas Schick]], \emph{$L^2$-index, KK-theory, and connections}, New York J. Math. 11 (2005) (\href{http://arxiv.org/abs/math/0306171}{arXiv:math/0306171}) \end{itemize} A standard textbook account in the context of [[KK-theory]] is in section 24.1 of \begin{itemize}% \item [[Bruce Blackadar]], \emph{[[K-Theory for Operator Algebras]]} \end{itemize} [[!redirects indices]] [[!redirects index theory]] [[!redirects index theories]] \end{document}