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Title Geometry of Dirac Operators
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Publication Date
Date Accessioned
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
Date Indexed 2020-07-20
Issued Date 2016-07-05 00:00:00

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