Generalized action-angle coordinates in toric contact spaces

Mirror Symmetry for Toric Branes on Compact Hypersurfaces

We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs. We define a linear sigma model for the brane geometry and describe a correspondence between dual toric polyhedra and toric brane geometries. The method is applied to study examples with obstructed and classically unobstructed brane moduli at various p…

Moduli Spaces of Contact Instantons

We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric $5$-manifolds and initiate the study of their structure. In the $K$-contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is given by the index of an operator elliptic transverse to the Reeb foliation. The moduli spaces are shown to be Kähler when the $5$-manifold $M$ is Sasakian and hyperKähler when $M$ is transverse Calabi-Yau. We show how the transverse index can be computed in various cases, in pa…

Brane quantization of toric Poisson varieties

In this paper we propose a noncommutative generalization of the relationship between compact Kähler manifolds and complex projective algebraic varieties. Beginning with a prequantized Kähler structure, we use a holomorphic Poisson tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented f…

瞬子、单极和复曲面HyperKaehler流形

Instantons, Monopoles and Toric HyperKaehler Manifolds

本文构造了k=1 SU（n）周期瞬子（或称caloron）的模空间上的度规.度规是复曲面超Kaehler的，其形式由Lee和Yi证明。环面坐标描述了剩余的U（1）^{n-1}规范不变性和Caloron的时间位置，也可以看作是构成Caloron的n个单极子的相位。（1，1，..，1)得到的热焓是卡隆的极限。计算是在Nahm数据的空间上进行的，这是合理的证明所考虑的情况下的Nahm结构的等距属性。另一种建设使用hyperKaehler商。无质量单极子的影响进行了简要讨论。

Sasaki流形和环面Sasaki-Einstein流形的横Kähler几何

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

本文从横Kähler几何的角度研究紧Sasaki流形，并将Kähler几何中的一些结果推广到Sasaki流形.特别是我们定义的积分不变量，阻碍存在的横向Kähler度量与调和陈形式。对于第一类Chern类情形，积分不变量$f_1$成为常数量曲率的横截Kähler度量存在的障碍.本文证明了紧致复曲面Sasaki流形上的横Kähler-Ricci孤子（或{ \it Sasaki-Ricci孤子}）的存在性，其中Reeb叶状的法丛的基本第一Chern形式为正，接触丛的第一Chern类为平凡.我们将进一步证明，如果$S$是一个紧复曲面Sasaki流形，在上述假设下，通过变形Reeb场，我们得到$S$上的Sasaki-Einstein结构.作为应用，我们得到了复射影平面两点爆破的典范丛的幂的单位圆丛上的非正则环面Sasaki-Einstein度量.

Counting Orbifolds

We present several methods of counting the orbifolds C^D/Gamma. A correspondence between counting orbifold actions on C^D, brane tilings, and toric diagrams in D-1 dimensions is drawn. Barycentric coordinates and scaling mechanisms are introduced to characterize lattice simplices as toric diagrams. We count orbifolds of C^3, C^4, C^5, C^6 and C^7. Some remarks are made on closed form formulas for the partition function that counts distinct orbifold actions.

N = 1特殊几何、混合Hodge变分与复曲面几何

N=1 Special Geometry, Mixed Hodge Variations and Toric Geometry

本文研究了一类N = 1的超对称II型紧化的超势。我们表明，它有一个重要的二维意义的手征环的拓扑扭曲理论的世界片。在开闭弦B模型中，这个手征环同构于某个相对上同调群V，这是处理开弦扇区和闭弦扇区的适当数学概念。V上的混合霍奇结构族则意味着超势具有某种几何结构。这种结构代表了一个全纯的，N = 1超对称推广的著名的N = 2特殊几何。它定义了开-闭B模型拓扑族上的一个可积联络，以及N = 1手征多重态中vev的空间\cal M上的一组特殊坐标.我们表明，它可以给出一个非常具体和简单的实现线性西格玛模型，这导致了一个强大的和系统的方法来计算精确的非微扰N = 1超势的一个广泛的类环面D-膜几何。

箭图表示的稳定性与拓扑变化

Stability of Quiver Representations and Topology Change

本文基于模表示的稳定性研究了轨道折叠C^3/Z_2 \times Z_2上D 0-膜模空间的相结构。从使用复曲面几何形状的分析中可知，该模型具有由触发器转换连接的多个相位。通过比较这两种方法的结果，我们得到了复曲面分解的几何形状和复曲面表示之间的对应关系。它示出的D-膜模空间，这是一个关键成分的消失的非几何相位的复曲面描述中产生的坐标的冗余，理解从monodromy周围的orbifold点。我们还从D ~ 0-膜的稳定性的角度讨论了为什么只有几何相出现。