On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$

We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contactomorphism group. To show that two toric contact structures with the same first Chern class are contact inequivalent, we use Morse-Bott contact homology. We treat a subclass of contact structures which include the Sasaki-Einstein contact structures $Y^{p,q}$ studied by physicists. In this subcase we give a complete solution to the contact equivalence problem by showing that $Y^{p,q}$ and $Y^{p'q'}$ are inequivalent as contact structures if and only if $p\neq p'$.Comment: 61 page

On the Conley-Zehnder index and Sasaki-Einstein manifolds

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2019. 2. Koert, Otto van .제 2장에서는 저자가 서울대학교 수리과학부에서 학위를 하는 동안 출판한 논문에 대해 소개하였습니다. 구체적으로 Reeb 벡터장을 사교공간상의 경로로 간주하고 그 Conley-Zehnder 지표와 몫공간으로서 생성된 기저 공간의 orbifold 천(Chern) 특성류 사이의 관계를 규명하였습니다. 이렇게 얻어진 관계를 우리에게 매우 익숙한 기본적인 예제들에 적용시켜 구체적인 값을 구하였습니다. 제 3장은 저자가 학위기간 동안 주로 연구한 분야인 사사키-아인슈타인 기하(Sasaki-Einstein geometry)에 대한 조사 보고서입니다. 기본적인 정의, 정리부터 흥미로운 예제, 존재성에 대한 걸림돌 이론(obstruction theory)등에 대해서 살펴보았습니다.In the second chapter, we prove a useful relation between the Conley-Zehnder indices of the Reeb vector flow action along periodic orbits in prequantization bundles and the orbifold Chern class of the base symplectic orbifolds motivated by the well-known case of manifolds. We also apply this method to primary examples. In the third chapter, we survey interesting properties on Sasaki-Einstein geometry from the elementary definitions and theorems to well-known examples and simple obstructions.Abstract i 1 Introduction 1 2 The Conley-Zehnder indices of the Reeb flow action along S1-fibers over certain orbifolds 4 2.1 The Conley-Zehnder index . . . . . . . . . . . . . . . . . . . . 4 2.1.1 The Maslov index . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 The Conley-Zehnder index . . . . . . . . . . . . . . . . 6 2.1.3 The Robbin-Salamon index . . . . . . . . . . . . . . . 7 2.2 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Classifying spaces . . . . . . . . . . . . . . . . . . . . . 12 2.3 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 The Boothby-Wang fibration . . . . . . . . . . . . . . . 15 2.3.2 The main theorem . . . . . . . . . . . . . . . . . . . . 16 2.3.3 The weighted projective spaces and their complete intersections . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.4 Some computations for non-principal orbits . . . . . . 30 2.3.5 Inertia orbifolds . . . . . . . . . . . . . . . . . . . . . . 32 3 A survey on Sasaki-Einstein manifolds 35 3.1 Sasakian structures and Einstein metrics . . . . . . . . . . . . 35 3.1.1 Symplectic manifolds and contact structures . . . . . . 35 3.1.2 Almost contact structures and Sasakian structures . . . 40 3.1.3 General relativity, Einstein manifolds . . . . . . . . . . 45 3.2 Kahler-Einstein metrics . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 Einstein conditions in Kahler metrics . . . . . . . . . . 51 3.2.2 Calabi conjecture and Calabi-Yau manifolds . . . . . . 54 3.2.3 Kahler-Einstein metrics on del Pezzo surfaces . . . . . 57 3.3 Sasaki-Einstein manifolds . . . . . . . . . . . . . . . . . . . . 62 3.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . 62 3.3.2 Toric Sasaki-Einstein manifolds . . . . . . . . . . . . . 66 3.3.3 Sasaki-Einstein metrics on Y pq . . . . . . . . . . . . . 75 3.3.4 Simple obstructions . . . . . . . . . . . . . . . . . . . . 80 Abstract (in Korean) 88Docto

Transversality, old and new

Resum: Estudi de la transversalitat, una eina molt útil de la topologia diferencial tant en varietats (de manera geomètrica) i els espais de jets. També es fa un breu repàs a la geometria diferencial necessària. Per últim es mostra un aplicació a l'estudi d'equacions diferencials en el tor, una varietat molt coneguda. Summary: In this TFG we study transversality, a very useful tool in Differential Topology, which is applied to manifolds (in a geometric way) and to the space of jets. A summary of basic facts in differential geometry is also included. Finally, we include an application to the study of differential equations on the torus, a very well-known manifold

Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory

In this thesis, we report on results in non-anticommutative field theory and twistor string theory, trying to be self-contained. We first review the construction of non-anticommutative N=4 super Yang-Mills theory and discuss a Drinfeld-twist which allows to regain a twisted supersymmetry in the non-anticommutative setting. This symmetry then leads to twisted chiral rings and supersymmetric Ward-Takahashi identities, which, when combined with the usual naturalness argument by Seiberg, could yield non-renormalization theorems for non-anticommutative field theories. The major part of this thesis consists of a discussion of various geometric aspects of the Penrose-Ward transform. We present in detail the case of N=4 super Yang-Mills theory and its self-dual truncation. Furthermore, we study reductions of the supertwistor space to exotic supermanifolds having even nilpotent dimensions as well as dimensional reductions to mini-supertwistor and mini-superambitwistor spaces. Eventually, we present two pairs of matrix models in the context of twistor string theory, and find a relation between the ADHM- and Nahm-constructions and topological D-brane configurations.Comment: PhD thesis, 280 pages, 9 figure