Special'Holonomy Group and Supersymmetric Cycle

点击翻译按钮获取中文摘要

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.