Geometry of Dirac Operators
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Authors
Beheshti Vadeqan, Babak
Date
2016-07-05
Type
thesis
Language
eng
Keyword
Atiyah-Singer Index Theorem , Dirac Operators , Elliptic Geometry
Alternative Title
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.
Description
Thesis (Master, Mathematics & Statistics) -- Queen's University, 2016-07-04 20:27:20.386
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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.
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.