Geometry of manifold of special holonomy and gauge theory/gravity correspondence

超弦理论中的d膜为研究规范理论的数学结构提供了一种新的方法。例如，特殊完整群G_2，自旋(7)流形中的d膜描述了超对称规范理论的非微扰动力学。与Y.Yasui合作，构造了具有SU(3)/U(1)主轨道的自旋(7)完整的显式度量。特别地，我们发现了一种新的度量，它具有一个半径有限的S^1的渐近圆。这是对四维引力瞬间的高维模拟；Taub-NUT和Atiyah-Hitchin。我们期望这些度量在M理论紧化中有一些应用。弦理论的目标之一是规范论与引力的统一，规范论与引力的对应关系近年来受到了广泛的关注。当我们考虑拓扑规范/弦理论时，这种对应关系可以用更严格的数学方法来表述。涅克拉索夫在四维规范理论中的瞬时计数是近年来的一项重要成就。我们与T.Eguchi合作，证明了瞬态计数的Nekrasov配分函数可以复制为拓扑弦理论的配分函数。考虑目标空间为A_{N-1} / P^1的ALE振动的拓扑弦，得到SU(N) Seiberg-Witten理论的配分函数。物质场可以通过在目标空间上爆炸来合并。这种计算方法与数学的各个分支有密切的关系，如表示论、组合论、结论和可积系统。

D-branes in superstring theory provide a new method of investigating the mathematical structure of gauge theory. For example, D-branes in manifolds of exceptional holonomy group G_2,Spin(7) describe non-perturbative dynamics of supersymmetric gauge theory. In collaboration with Y.Yasui, we constructed explicit metrics of Spin(7) holonomy, that are of cohomogeneity one with SU(3)/U(1) principal orbit. In particular we found a new type of metrics that has asymptotically a circle S^1 of finite radius. This is a higher dimensional analog of four dimensional gravitational instantons ; Taub-NUT metic and Atiyah-Hitchin metic. We expect these metrics have some applications to M theory compactification.One of aims of string theory is a unification of gauge theory and gravity and gauge theory/gravity correspondence has been attracted much attention recently. When we consider topological gauge/string theory, the correspondence can be formulated mathematically more rigorous way. One of recent important achievements is instanton counting in four dimensional gauge theory by Nekrasov. In collaboration with T.Eguchi, we show that Nekrasov's partition function of instanton counting can be reproduced as a partition function of topological string theory. The partition function for SU(N) Seiberg-Witten theory is obtained by considering topological string whose target space is the ALE fibration of type A_{N-1} over P^1. Matter field can be incorporated by making a blow up on the target space. The methods used in this computation have close relations to various branches in mathematics, such as representation theory, combinatorics, knot theory and integrable systems.