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Special'Holonomy Group and Supersymmetric Cycle

特殊完整群和超对称循环

基本信息

批准号：
12640074
负责人：
KANNO Hiroaki
金额：
$ 1.86万
依托单位：
Nagoya University (2001)Hiroshima University (2000)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

Mirror symmetry is one of important topics in' the geometry of manifold of special holonomy.We have investigated five dimensional supersymmetric gauge theories using the principle of local mirror symmetry. Based on the Hirzebruch surface Fa and its blow ups at N(< 5) points, we obtain a family of elliptic curves, from which the prepotential of five dimensional supersymmetric gauge theory compactified on S1 can be derived. Our results implies a new insight into the instanton expansion of the prepotential. It is expected that five dimensional supersymmetric gauge theories are deeply related to the geometry of rational elliptic surface and the theory of simple elliptic singularities.We have also discussed deformations of our Spin(7] metrics within a formal power series expansion. Using a metric ansatz of cohomogeneity one with the principal orbit SU(S)/U(l), we have found new explicit metrics of Spin(7) holonomy. They are expected to describe local geometry of an isolated conical singularity which is developing when a SUSY 4-cycle CP2 shrinks,in a Spin(7} manifold. Our new metric is asymptotically conical in the sense that asymptotically there is a circle S1 with a finite radius, which is important from the viewpoint of M theory.We have also discussed deformations of our Spin(7] metrics within a formal power series expansion. Hence they are regarded as a higher dimensional analog of Taub-NUT metric and Atiyah-Hitchin metric in four dimensions. We hope these metrics have applications to M theory compactification.

镜像对称是特殊完整流形几何中的重要课题之一。我们利用局域镜像对称原理研究了五维超对称规范理论。基于Hirzebruch曲面Fa及其在N(< 5)点处的爆炸，我们得到一族椭圆曲线，由此可以推导出在S1上压缩的五维超对称规范理论的预势。我们的结果意味着对预势瞬时展开的新见解。预计五维超对称规范理论与有理椭圆面几何和简单椭圆奇点理论密切相关。我们还讨论了形式幂级数展开中 Spin(7) 度量的变形。使用与主轨道 SU(S)/U(l) 共齐的度量拟设，我们发现了 Spin(7) 新的显式度量和乐性。他们期望描述在 Spin(7} 流形中 SUSY 4 周期 CP2 收缩时形成的孤立圆锥奇点的局部几何。我们的新度量是渐近圆锥的，因为渐进地存在一个具有有限半径的圆 S1，这从 M 理论的角度来看很重要。我们还讨论了形式幂级数中 Spin(7] 度量的变形扩张。因此，它们被视为 Taub-NUT 度量和 Atiyah-Hitchin 度量在四个维度上的高维模拟。我们希望这些指标能够应用于 M 理论压缩。