Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe

In three spacetime dimensions, general relativity drastically simplifies, becoming a ``topological'' theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2+1)-dimensional vacuum gravity in the setting of a spatially closed universe.Comment: 61 pages, draft of review for Living Reviews; comments, criticisms, additions, missing references welcome; v2: minor changes, added reference

Invertible sheaves on generic rational surfaces and a conjecture of Hirschowitz's.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (p. 63-68).Ph.D

오비다양체의 사교기하와 디오판투스 방정식

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2015. 2. 조철현.We study symplectic geometry of orbifolds, especially which are necessary to extend the Lagrangian intersection Floer theory to the one of orbifold setting. First, we give another definition of the Maslov indices of bundle pairs via curvature integral of L-orthogonal unitary connection. This definition naturally extends to the one of orbi-bundle pairs with interior singularities. Secondly, we investigate the notion of orbifold embedding. When the target orbifold is a global quotient of a smooth manifold by the action of a Lie group G, we show that orbifold embeddings are equivariant with G-equivariant immersions. In the last part of the dissertation, we compute quantum cohomology of elliptic P1 orbifolds via classifying holomorphic orbi-spheres in those orbifolds. Interestingly, we find that these orbi-spheres have an one-to-one correspondence with the solutions of certain Diophantine equations depending on the lattice structures on the universal covers of elliptic P1 orbifolds constructed from the preimages of three singular points.1 Introduction 1 2 Preliminaries 7 2.1 Symplectic geometry 7 2.2 Orbifolds 9 2.3 Orbifold fundamental group 23 2.4 Orbifold covering theory 24 2.5 Orbifold Gromov-Witten theory 25 3 Maslov index via Chern-Weil theory and its orbifold analogue 36 3.1 Maslov index via orthogonal connection 36 3.2 Equivalence of two Maslov indices 40 3.3 Properties of Chern-Weil Maslov index 46 3.4 The case of transversely intersecting Lagrangian submanifolds 49 3.5 Orbifold Maslov Index 55 4 On orbifold embeddings 60 4.1 Orbifold embeddings 60 4.2 Inertia orbifolds and orbifold embeddings 66 4.3 Orbifold embeddings and equivariant immersions 70 4.4 Construction of equivariant immersions from orbifold embeddings 77 4.5 General case 82 5 Holomorphic orbi-spheres in elliptic P1 orbifolds and Diophantine equations 88 5.1 Orbi-maps between two dimensional orbifolds 88 5.2 Holomorphic orbifold maps 91 5.3 The quantum cohomology ring of P13,3,3 99 5.4 Further applications:(2,3,6),(2,4,4) 107 5.5 Theta series 122Docto

복소, 무질서 및 광학적 비선형 퍼텐셜에서의 대칭성 붕괴를 통한 빛의 흐름 제어

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2015. 8. 박남규.매질 내 빛의 흐름은 통상적으로 거시적 맥스웰 방정식에 의해 정의된다. 동질성 및 등방성을 가지고, 선형적이며, 시간에 대해 일정한 광학 매질 변수를 갖는 이상적인 매질에서는 광파의 양상이 페르마의 원리의 직접적인 예인 진동하는 전자기장의 직진 형태로, 간단하며 직관적이다. 이러한 평면파적 특성은 기하 광학의 바탕이며, 슈뢰딩거 방정식 형태의 파동 방정식이 갖는 다양한 대칭성 (병진 대칭, 키랄 대칭, 에르미트 대칭, 로렌츠 대칭 및 시간 반전 대칭)의 보존에서 그 원리을 찾을 수 있다. 렌즈, 거울 및 프리즘과 같은 고전적인 방식에서조차, 빛의 흐름을 조절키 위해서는 일부 광학적 대칭성의 붕괴를 필요로 한다. 비균질 매질에서의 병진 대칭의 붕괴는 굴절, 반사, 회절과 같은 산란 기반 빛 제어를 위한 고전적인 방법이다. 전파 시의 빛 에너지의 소모 또는 증폭은 파동 방정식의 비에르미트 헤밀토니안에 의해 정량화된다. 키랄 분자로 이루어진 매질은 광학 활성, 즉 빛의 편광을 돌릴 수 있도록 한다. 천문학에서 별 및 은하 움직임의 관찰에 이용되는 광학적 도플러 효과는 로렌츠 대칭성을 붕괴시키는 광원의 시간에 따른 변화에 기반한다. 비직관적인 이론적 결과물 및 향상된 공정 기술을 포함하는 광학 분야의 최근 성과들은 이제 비고전적인 빛의 흐름을 이끌어내기 위한 광학적 퍼텐셜 제어의 새로운 영역을 개척하고 있다. 메타 물질 개념과 연계된 나노 스케일 기술은 단방향 빛 전파, 변형된 스넬의 법칙, 음굴절율, 투명 망토, 완전 흡수체와 같은 특이한 빛의 흐름을 지원하는, 이론적으로 증명된 인조 매질의 설계를 가능케 한다. 광 증폭 기술의 발전은 양자역학적 개념인 패리티-시간 대칭성의 구현에 적용되어, 복소 퍼텐셜에서의 새로운 종류의 광학을 탄생시켰다. 이러한 성취물들은 맥스웰 방정식에서의 더 넓고 급격한 형태의 대칭성 붕괴에 기반하기 때문에, 의도된 빛의 흐름 조절을 위해서는 다양한 대칭성 붕괴에 관한 심도있는 연구가 필요하다. 본 학위 논문에서는 복소, 불규칙, 비선형 광학 퍼텐셜과 같은 다양한 플랫폼에서의 대칭성 붕괴에 대하여 살펴보고자 한다. 본 연구는 패리티-시간 대칭성, 키랄 특성, 인과율, 초대칭, 생물 모방 기술, 모드 경계 광학 및 느린 빛 원리와 연계된 빛의 특이한 흐름에 집중한다. 본 연구진이 이끌어낸 비직관적인 개념 및 광소자의 새로운 설계 기법 관련 결과들은 비고전적인 빛의 흐름에 기반한 미래 광학 발전에 도움이 될 것이다.The flow of light in matters is usually defined by macroscopic Maxwells equations. In ideal media with homogeneous, isotropic, linear, and time-invariant optical material parameters, the aspect of light wave dynamics is simple and intuitive: propagating straight with oscillated electromagnetic fields, as the direct example of Fermats principle. This planewave dynamics, the basis of geometric optics, originates from the conservation of various symmetries of the Schrodinger-like wave equation, including translational and chiral symmetry, Hermitian symmetry, Lorentz reciprocity, and time-reversal symmetry. To control the flow of light even in a classical manner such as lens, mirror, and prism, some parts of the symmetries in optics should be broken. Breaking the translational symmetry with inhomogeneous materials is the traditional method of controlling light by scattering such as refraction, reflection, and diffraction. The dissipation or amplification of optical energy during the propagation is quantified by the non-Hermitian Hamiltonian of the wave equation. The materials composed of chiral molecules allow the rotation of the polarization of light, i.e. optical activity. The optical Doppler effect, which has been employed in astronomy for the observation of the motion of stars and galaxies, is based on the time-varying position of light sources, breaking Lorentz reciprocity. Recent achievements in optics, including counterintuitive theoretical results and improved fabrication technologies, have now been pioneering unprecedented regimes of controlling optical potentials which derive non-classical flow of light. Nano-scale technologies linked with the concept of metamaterials have opened a path to the design of theoretically-demonstrated artificial media supporting extraordinary light flows: such as unidirectional light flow, modified Snells law, negative index, cloaking, and perfect absorption. The development of optical amplification techniques has been applied to the realization of the quantum-mechanical notion of parity-time symmetry: stimulating a new class of optics in complex potentials. Because these achievements have been based on broader and drastic forms of symmetry breaking in Maxwells equations, in-depth investigation of various symmetry breakings is now imperative to realize designer manipulation of light flow. In this dissertation, we explore symmetry breakings in various platforms: complex, disordered, and nonlinear optical potentials. The investigation is focused on unconventional flows of light linked with the notions of parity-time symmetry, chirality, causality, supersymmetry, biomimetics, mode junction photonics, and slow-light. We believe that our results including counterintuitive concepts and novel design methods for optical devices will be the foundation of future development in optics based on non-classical flow of light.Table of Contents Abstract i Table of Contents iv List of Figures viii Chapter 1 Introduction １ 1.1 Why should we break the symmetry of light? １ 1.2 Outline of the dissertation ２ Chapter 2 Parity-Time Symmetric Optics ４ 2.1 Introduction to PT-symmetric optics ５ 2.2 PT-symmetric waves in the spatial domain １１ 2.2.1 2-level chirped system １１ 2.2.2 N-level photonic molecule ２４ 2.3 PT-symmetric waves in momentum domains ４３ 2.3.1 Optical chirality in low-dimensional eigensystems ４４ 2.3.2 Interpretation of PT-symmetry in k-space ６３ 2.4 Conclusion ７５ Chapter 3 Disordered Optics ７６ 3.1 Introduction to disordered optics ７７ 3.2 Supersymmetric bandgap in disorder ７８ 3.2.1 Wave dynamics in random-walk potentials ７９ 3.2.2 Supersymmetric transformation for isospectrality ８３ 3.2.3 Bloch-wave family with tunable disorder ８６ 3.3 Biomimetic disordered surface ９１ 3.4 Conclusion ９８ Chapter 4 All-Optical Devices with Nonlinearity ９９ 4.1 Introduction to all-optical devices １００ 4.2 Mode junction photonics １０１ 4.2.1 Photonic Junction Diode １０５ 4.2.2 Multi-Junction Half Adder １１３ 4.3 Slow-light enhanced optical functionalities １１５ 4.3.1 Multiband slow light １１６ 4.3.2 Optical A/D converter １２６ 4.3.3 All-optical A/D converter １３７ 4.3.4 Travelling-wave all-optical isolator １４３ 4.4 Conclusion １４９ Chapter 5 Conclusion １５０ Appendix A Eigenvalues in PT-Meta-molecules １５２ Appendix B Supplements for Section 2.3.1 １５７ B.1 Planewave solution of a PT-symmetric optical material １５７ B.2 Density of optical chirality for complex eigenmodes １５８ B.3 Effect of imperfect PT symmetry on the modal chirality １５９ B.3.1 Broken symmetry in the real part of permittivity １５９ B.3.2 Broken anti-symmetry in the imaginary part of the permittivity １６１ B.4 Transfer between RCP and LCP modes in the PT-symmetric chiral material １６２ B.4.1 Propagation of complex eigenmodes １６２ B.4.2 Strength of chiral conversion CCS before the EP １６３ B.5 The state of polarization (SOP) at the EP: Optical spin black hole １６４ B.6 Giant chiral conversion in the resonant structure １６５ B.7 Detailed information of fabrication and experiment in THz chiral polar metamaterials １６６ B.7.1 Fabrication process of THz chiral polar metamaterials １６６ B.7.2 THz-TDS system for the measurement of intermodal chirality １６７ B.8 Realization of PT-symmetric permittivity in metamaterial platforms １６７ B.9 Design parameters of chiral waveguides １７１ B.10 Low-dimensional linear polarization １７１ Appendix C Detailed Derivation for Section 2.3.2 １７３ C.1 Detailed derivation of Eq. (2.20) １７３ C.2 Serial calculation of discretized coupled mode equations １７５ Appendix D Analytical Methods for Section 3.2 １７７ D.1 Details of the FDM and FGH method １７７ D.2 Calculation of the Hurst exponent １７７ Appendix E Supplements for Section 4.2 １７９ E.1 Details of the device structures and numerical method used in the study １７９ E.2 Coupled mode theory for the di-atomic photonic junction diode １８１ E.2.1 Analytical model and coupled mode equations １８１ E.2.2. Solution of resonator field (a1, a2, a3) １８３ E.2.3 Implementation of Kerr nonlinearity and calculation of diode throughput １８５ Bibliography １８７ Abstract in Korean ２０３Docto

Resolving the QCD phase structure

This thesis discusses the quantitative description of the phase structure of Quantum Chromo- dynamics (QCD). We find that, in strongly correlated theories such as QCD, even a qualitative investigation of the phase structure can require highly quantitative methods. Hence, the de- velopment of a method with systematic error control is essential. In the present work, we use functional renormalisation group (fRG) method to this aim. This work focusses on three ideas: Firstly, we identify quantitatively dominating and sub-leading scattering-processes in our approximations. This allows a formulation of low energy effective theories of the four-quark interaction, as well as the description of gluon condensation. For the former, we present results for meson and quark masses. The latter provides an estimate of the Yang-Mills mass gap. Secondly, we further develop the use of highly precise numerical methods from fluid-dynamics in the fRG. In particular we use Discontinuous Galerkin methods, which are able to capture shock-development. Shock-waves are found to play a big role in a possible creation-mechanism of first-order phase transitions. Lastly, we focus on general RG-transformations (gRGt). For example, they allow a real time formulation of fRG flows and hence give access to spectral functions. Furthermore, we use them to formulate complex RG-flows, which enables us to locate Lee-Yang singularities in the complex plane and extrapolate the position of (real) phase transitions. Finally, we also use gRGts to formulate significant qualitative improvements of current fRG approximation schemes by means of dynamical field transformations

Sealing Potential of Shale Sequences through Seismic Anisotropy Analysis

This study investigates the potential relation of seismic anisotropy measured by surface seismic and the sealing potential of the shale sequences. Two case studies analyzed such a relationship. The Gippsland basin and Exmouth sub-basin are both hosts to prolific hydrocarbon resources and offer plenty of seismic data and sealing potential measurements. Weak anisotropy parameters extracted from carefully reprocessed seismic data show in both cases a positive correlation between sealing capacity and anisotropy of the shale