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 Author Beheshti Vadeqan, Babak Title Geometry of Dirac Operators URL http://hdl.handle.net/1974/14633 Publication Date 2016 Date Accessioned 2016-07-05 15:45:51 Discipline/Department Mathematics and Statistics University/Publisher Queens University Abstract Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then $\textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V})$ where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes. Subjects/Keywords Atiyah-Singer Index Theorem ; Dirac Operators ; Elliptic Geometry Language en Rights Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada ProQuest PhD and Master's Theses International Dissemination Agreement Intellectual Property Guidelines at Queen's University Copying and Preserving Your Thesis Creative Commons - Attribution - CC BY This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner. This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner. [Always confirm rights and permissions with the source record.] Country of Publication ca Record ID handle:1974/14633 Repository queens Date Retrieved 2020-07-15 Date Indexed 2020-07-20 Issued Date 2016-07-05 00:00:00

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operators and specifically Dirac operators contain certain information about the geometry and topology of the base manifold and the related vector bundles. In particular, using the heat kernel, we aim to prove the celebrated Atiyah-Singer index theorem for…

Dirac operators. We shall also see that the study of Dirac operators reveals some interconnections between the geometry and the topology of the underlying manifold. Perhaps one of the most well-known results of this type is the Gauss-Bonnet theorem: 1 χ…

…x5D; and Atiyah-Bott-Patodi [1] improved the result for a more general type of elliptic differential operators which are said to be Dirac type operators. Let us put these ideas in the context of elliptic operators. Let P : C ∞ ( E)…

…Patodi [1] that the characteristic polynomial on the right hand side is Â( TM ) and 10 we have /+ = Ind D Z M Â( TM) The above operators are all typical of a large class of elliptic operators called Dirac operators

…In this thesis we are planning to study this type of operators. We are going in detail through the algebraic and geometric structures required to construct Dirac operators on manifolds. We will start by introducing Clifford algebras and their…

…representation which is the algebraic foundation of Dirac operators structure. In chapter 3 we introduce Clifford bundles and Dirac operators. This is in fact the geometric facet of our constructions. Then we study the analysis of Dirac including the heat…

…equation and the asymptotic expansion of the heat kernel. In the last chapter we go through Getzler’s" idea to prove the index theorem and we will see that Equation 1.4 for Dirac operators becomes Ind D = Z M Â( TM) ∧ ch(V )…

…x28;1.5) the Atiyah-Singer index theorem. This is also equivalent to Theorem 1.1 for the case of Dirac operators on Clifford bundles. We assume as background basics of differential geometry in particular Chern-Weyl theory of characteristic classes…