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Representation of Geometric Objects by Path Integrals

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Title: Representation of Geometric Objects by Path Integrals
Other Titles: 経路積分による幾何的対象の表現
Authors: 桑田, 健1 Browse this author
Authors(alt): Kuwata, Ken1
Issue Date: 24-Mar-2022
Publisher: Hokkaido University
Abstract: In this thesis, we study two problems by using physical models with super-symmetry and their path integral. Since this thesis deals with these topics from a common perspective, but in different ways, we have divided this thesis into two parts. One is deriving fixed-point theorem using path integral [25]. The other is the Euler number and Mathai-Quillen formalism in the Grassmann manifold [20]. This thesis is based on [20], [25] and [28]. In the part I, we derive the Bott residue formula by using the topological sigma model (A-model) that describes dynamics of maps from CP1 to a Kähler manifold M, with potential terms induced from a holomorphic vector field K on M [25]. The Bott residue formula represents the intersection number of Chern classes of holomorphic vector bundles on M as the sum of contributions from fixed point sets of K on M. Our strategy is to represent the integral of differential form on M by a correlation function and show that the correlation function is obtained by collecting contributions from the zero set of K. It is realized by showing the invariance of correlation function for the parameter of potential terms. As an effect of adding a potential term to the topological sigma model, we are forced to modify the BRST symmetry of the original topological sigma model. In the part II, we provide a recipe for computing Euler number of Grassmann manifold G(k;N) by using Mathai-Quillen formalism [33] and Atiyah-Jeffrey construction [3]. Especially, we construct the path integral representation of Euler number of G(k;N) [20]. As a by-product, we construct free fermion realization of cohomology ring of G(k;N). It means that the cohomology ring of G(k;N) can be represented by fermionic fields that appear in our model. As an application, we calculate some integrals of cohomology classes by using fermion integrals [28].
Conffering University: 北海道大学
Degree Report Number: 甲第14776号
Degree Level: 博士
Degree Discipline: 理学
Examination Committee Members: (主査) 教授 石川 剛郎, 教授 吉永 正彦, 教授 泰泉寺 雅夫 (岡山大学大学院自然科学研究科)
Degree Affiliation: 理学院(数学専攻)
Type: theses (doctoral)
Appears in Collections:課程博士 (Doctorate by way of Advanced Course) > 理学院(Graduate School of Science)
学位論文 (Theses) > 博士 (理学)

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