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geometry</a></p> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+stable+homotopy+theory">noncommutative stable homotopy theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/genus">genus</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></strong></p> <h2 id="definitions">Definitions</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+module">Hilbert module</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a>, <a class="existingWikiWord" href="/nlab/show/pseudodifferential+operator">pseudodifferential operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symbol+of+a+differential+operator">symbol of a differential operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+operator">elliptic operator</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+complex">elliptic complex</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a></li> </ul> </li> </ul> <h2 id="index_theorems">Index theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+index">topological index</a>, <a class="existingWikiWord" href="/nlab/show/analytical+index">analytical index</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Singer+index+theorem">Atiyah-Singer index theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gauss-Bonnet+theorem">Gauss-Bonnet theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hirzebruch-Riemann-Roch+theorem">Hirzebruch-Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+analytic+index+theory">global analytic index theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hirzebruch+signature+theorem">Hirzebruch signature theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mishchenko-Fomenko+index+theorem">Mishchenko-Fomenko index theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baum-Connes+conjecture">Baum-Connes conjecture</a></p> </li> </ul> <h2 id="higher_genera">Higher genera</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#statement'>Statement</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesPhysics'>From the point of view of physics</a></li> </ul> </ul> </div> <h2 id="statement">Statement</h2> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a> and <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a> associated two differently defined numbers to an <a class="existingWikiWord" href="/nlab/show/elliptic+operator">elliptic operator</a> on a manifold: the <strong><a class="existingWikiWord" href="/nlab/show/topological+index">topological index</a></strong> and the <strong><a class="existingWikiWord" href="/nlab/show/analytical+index">analytical index</a></strong>. The index theorem asserts that the two are equal.</p> <p>The index theorem generalizes earlier results such as the <a class="existingWikiWord" href="/nlab/show/Riemann-Roch+theorem">Riemann-Roch theorem</a>.</p> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The original articles:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a>, <em>The index of elliptic operators I</em>, Ann. of Math. (2) <strong>87</strong> (1968) pp. 484–530; <em>III</em>, Ann. of Math. (2) <strong>87</strong> (1968) pp. 546–604; <em>IV</em>, Ann. of Math. (2) <strong>93</strong> (1971) pp. 119–138; <em>V</em>, Ann. of Math. (2) <strong>93</strong> (1971) pp. 139–149</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>The index of elliptic operators II</em>, Ann. of Math. (2) <strong>87</strong> (1968) pp. 531–545</p> </li> </ul> <p>Review:</p> <ul> <li id="HirzebruchGergerJung92"> <p><a class="existingWikiWord" href="/nlab/show/Friedrich+Hirzebruch">Friedrich Hirzebruch</a>, Thomas Berger, Rainer Jung, chapter 6 of: <em>Manifolds and Modular Forms</em>, Aspects of Mathematics <strong>20</strong>, Viehweg (1992), Springer (1994) [<a href="https://doi.org/10.1007/978-3-663-10726-2">doi:10.1007/978-3-663-10726-2</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf">pdf</a>]</p> </li> <li id="Gilkey95"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Gilkey">Peter Gilkey</a>, Section 5.2 of: <em>Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem</em>, 1995 (<a href="https://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf">pdf</a>)</p> </li> <li> <p>Rafe Mazzeo, <em>The Atiyah-Singer Index theorem: what it is and why you should care</em>, (<a href="https://docplayer.net/52573131-The-atiyah-singer-index-theorem-what-it-is-and-why-you-should-care-rafe-mazzeo-october-10-2002.html">slides</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>The Atiyah-Singer index theorem</em>, (<a href="https://web.ma.utexas.edu/users/dafr/cmsa.pdf">slides</a>)</p> </li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem">Atiyah-Singer index theorem</a></em></li> </ul> <p>A proof of the Atiyah-Singer index theorem in terms of <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>/<a class="existingWikiWord" href="/nlab/show/E-theory">E-theory</a> has been given by <a class="existingWikiWord" href="/nlab/show/Nigel+Higson">Nigel Higson</a>, an account is in</p> <ul> <li id="HigsonRoe"><a class="existingWikiWord" href="/nlab/show/Nigel+Higson">Nigel Higson</a>, <a class="existingWikiWord" href="/nlab/show/John+Roe">John Roe</a>, <em>Lectures on operator K-theory and the Atiyah-Singer Index Theorem</em> (2004) (<a href="http://folk.uio.no/rognes/higson/Book.pdf">pdf</a>)</li> </ul> <p>A lightning review of the proof is on the last pages of</p> <ul> <li id="Introduction"><em>Introduction to KK-theory and E-theory</em>, Lecture notes (Lisbon 2009) (<a href="http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf">pdf slides</a>)</li> </ul> <h3 id="ReferencesPhysics">From the point of view of physics</h3> <p>The index theorem has an interpretation in terms of the <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> of the <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> on the given space. See also at <a href="https://ncatlab.org/nlab/show/supersymmetric+quantum+mechanics#RelationToIndexTheory">supersymmetric quantum mechanics</a>.</p> <p>Traditional physics arguments along these lines include for instance</p> <ul> <li> <p>P. Windey, <em>Supersymmetric quantum mechanics and the Atiyah–Singer index theorem</em>, Acta Physica Polonica, <strong>B15</strong> (1984). (<a href="http://th-www.if.uj.edu.pl/~acta/vol15/pdf/v15p0435.pdf">PDF</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Luis+Alvarez-Gaum%C3%A9">Luis Alvarez-Gaumé</a>, <em>Supersymmetry and the Atiyah-Singer index theorem</em>, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (<a href="http://projecteuclid.org/euclid.cmp/1103940278">EUCLID</a>)</p> </li> <li> <p>Florian Hanisch, Matthias Ludewig, <em>A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold</em>, (<a href="https://arxiv.org/abs/1709.10027">arXiv:1709.10027</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 9, 2023 at 18:27:02. 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