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Conformal Field Theory and Schramm-Loewner Evolution in A Doubly Connected Domain : 이중연결영역에서의 등각장론과 슈람-뢰브너 전개
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | 김판기 | - |
| dc.contributor.author | 변성수 | - |
| dc.date.accessioned | 2022-03-25T06:00:35Z | - |
| dc.date.available | 2022-03-25T06:00:35Z | - |
| dc.date.issued | 2021 | - |
| dc.identifier.other | 000000167125 | - |
| dc.identifier.uri | https://hdl.handle.net/10371/177547 | - |
| dc.identifier.uri | https://dcollection.snu.ac.kr/common/orgView/000000167125 | ko_KR |
| dc.description | 학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2021.8. 김판기. | - |
| dc.description.abstract | The aim of this thesis is to develop a constructive conformal field theory of Gaussian free field in a doubly connected domain and its implementations to the theory of annulus Schramm-Loewner evolution (SLE). In particular, we derive certain types of BPZ equations based on Eguchi-Ooguri's version of Ward's equations, which describe the Virasoro fields in terms of the Lie derivative operators and the Teichmüller modular parameter. As applications, we find Coulomb gas solutions to the null-vector equations for annulus SLE partition functions and construct a large class of martingale-observables for variants of annulus SLE processes. | - |
| dc.description.abstract | 본 학위논문의 목적은 이중연결영역에서의 가우스 자유장으로부터 건설된 등각장론과 이의 환 슈람-뢰브너 전개 (SLE) 이론에의 구현을 정립하는 것이다. 특히, 리 (Lie) 미분 연산자 및 타이히뮐러 (Teichmüller) 매개 변수를 통해 비라소로 (Virasoro) 장을 기술하는 에구치-오구리 (Eguchi-Ooguri) 버전의 와드 (Ward) 방정식들을 기반으로 특정 유형의 BPZ 방정식들을 도출한다. 이에 대한 응용으로서, 환 SLE 분배함수에 대한 영-벡터 방정식의 쿨롱 (Coulomb) 기체 해들을 찾고, 여러가지 환 SLE 들에 대한 마팅게일 관측들을 건설한다. | - |
| dc.description.tableofcontents | 1 Introduction and overview 1
2 Formulation of main results 7 2.1 Notation 7 2.2 Basic setup 8 2.3 Main results 10 2.4 Organization of the thesis 19 3 Gaussian free field 20 3.1 Special functions 20 3.1.1 Elliptic functions 20 3.1.2 Schottky-Klein prime function 28 3.2 Representation of Green's functions 30 3.2.1 Dirichlet and Excursion-reflected boundary conditions 30 3.2.2 Green's function in a doubly connected domain 33 3.2.3 Green's function in a multiply connected domain 34 3.2.4 Green's function on a compact Riemann surface 35 3.2.5 Riemann-Hilbert boundary conditions 37 3.3 Variants of GFFs 41 3.3.1 GFF in a multiply connected domain 41 3.3.2 GFF on a compact Riemann surface 42 3.3.3 Schottky double construction of GFF 43 4 Schramm-Loewner evolution 48 4.1 Loewner equations in a multiply connected domain 50 4.1.1 Standard domain and canonical mapping 50 4.1.2 Loewner vector field and flow 53 4.2 Evolutionary description of SLE 55 4.2.1 SLE in a simply connected domain 55 4.2.2 SLE in a doubly connected domain 56 5 Conformal Fock space fields 59 5.1 Correlation functions of Fock space fields 59 5.1.1 One parameter family of Gaussian free fields 59 5.1.2 Fock space fields 60 5.1.3 OPE calculations 61 5.2 Conformal geometry of Fock space fields 63 5.2.1 Lie derivative of conformal Fock space field 65 5.2.2 Stress energy tensor 65 5.3 Central charge modification 67 5.4 Formal bosonic fields and neutrality condition 70 5.5 Coulomb gas correlation function 72 5.6 Background charge modifications 75 6 Eguchi-Ooguri and Ward's equations 81 6.1 Ward's functional and Ward's identities 81 6.2 Eguchi-Ooguri equations 84 6.3 Ward's equations 96 6.4 Generalized Eguchi-Ooguri equations 97 7 Annulus SLE martingale-observables 102 7.1 One-leg operator and BPZ equations 102 7.2 Null-vector equations 108 7.3 BPZ-Cardy equations 111 7.4 Examples 114 7.4.1 Bosonic observable 114 7.4.2 Schramm's observable 117 7.4.3 Restriction observable 118 7.5 Non-atomic background charges 129 7.6 Periodization 132 7.7 Screening 135 7.7.1 Partition functions for two commuting SLEs 145 7.7.2 Crossing annulus SLE partition function 148 7.7.3 A classical limit 152 8 Martingale-observables for continuum interface curves 154 8.1 GFF observables 154 8.1.1 Hitting probabilities of the inner boundary component 156 8.1.2 Crossing case 159 8.1.3 GFF with general boundary data 159 8.1.4 Vertex observable 162 8.2 LERW observables 167 8.2.1 LERW partition functions 168 8.2.2 Generalized Poisson kernel 171 8.2.3 Ending point distribution of standard SLE2 172 8.2.4 Bosonic observables for LERW 173 8.3 Critical Ising observables 175 8.4 Annulus SLE with one marked point 177 8.4.1 Bosonic observables 178 8.4.2 Vertex observables for standard SLE 178 9 Ward's identities in random matrix theory 183 9.1 Global Ward's identities and their applications 183 9.2 Rescaled Ward's identities and their applications 188 Bibliography 191 Abstract (in Korean) 205 Acknowledgement (in Korean) 206 | - |
| dc.format.extent | v, 207 | - |
| dc.language.iso | eng | - |
| dc.publisher | 서울대학교 대학원 | - |
| dc.subject | Conformal field thoery | - |
| dc.subject | Schramm-Loewner evolution | - |
| dc.subject | Gaussian free field | - |
| dc.subject | doubly connected domain | - |
| dc.subject | partition functions | - |
| dc.subject | martingale-observables | - |
| dc.subject | 등각장론 | - |
| dc.subject | 슈람-뢰브러 전개 | - |
| dc.subject | 가우스 자유장 | - |
| dc.subject | 이중연결영역 | - |
| dc.subject | 분배함 수 | - |
| dc.subject | 마팅게일 관측 | - |
| dc.subject.ddc | 510 | - |
| dc.title | Conformal Field Theory and Schramm-Loewner Evolution in A Doubly Connected Domain | - |
| dc.title.alternative | 이중연결영역에서의 등각장론과 슈람-뢰브너 전개 | - |
| dc.type | Thesis | - |
| dc.type | Dissertation | - |
| dc.contributor.AlternativeAuthor | Sung-Soo Byun | - |
| dc.contributor.department | 자연과학대학 수리과학부 | - |
| dc.description.degree | 박사 | - |
| dc.date.awarded | 2021-08 | - |
| dc.identifier.uci | I804:11032-000000167125 | - |
| dc.identifier.holdings | 000000000046▲000000000053▲000000167125▲ | - |
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