Special'Holonomy Group and Supersymmetric Cycle

镜像对称是特殊完整流形几何中的一个重要课题。我们利用局部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）点处的膨胀，我们得到了一组椭圆曲线，由此可以导出在S1上紧化的五维超对称规范理论的预势。我们的结果意味着对预电位的瞬时膨胀有了新的认识。五维超对称规范理论与有理椭圆曲面几何和简单椭圆奇点理论有着密切的联系。我们还讨论了在正式幂级数展开中自旋(7)度量的变形。利用主轨道为SU(S)/U(l)的同质性度量，我们得到了自旋(7)完整度的新的显式度量。期望它们描述自旋(7)流形中SUSY 4环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，意思是渐近地存在一个半径有限的圆S1，这从M理论的观点来看是很重要的。我们还讨论了在正式幂级数展开中自旋(7)度量的变形。因此，它们被认为是四维中Taub-NUT度规和Atiyah-Hitchin度规的高维类比。我们希望这些度量可以应用于M理论的紧化。

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.