Geometry of Higher Dimensional Black Hole and Compact Einstein Space

我们旨在从黑洞的角度研究特殊完整的紧致爱因斯坦流形和黎曼流形。这些几何形状在物理上被解释为更高维度的引力瞬子，并且与弦理论所期望的新几何形状密切相关。特别是紧致Sasaki-Einstein几何在AdS/CFT对应中得到了广泛的研究。我们在球束上构造了一个新的爱因斯坦度量的无穷级数。这些指标表现为5维AdS Kerr黑洞的一定极限。在特殊情况下，度规给出了齐次Sasaki-Einstein度规。我们的构造是Page方法的高维版本，Page给出了第一个del Pezzo曲面上的非齐次爱因斯坦度规。近年来，人们从AdS/CFT对应的角度对环形Calabi-Yau锥尖端的D3膜进行了广泛的研究。为了探索非共形场理论，考虑锥度规的变形是很自然的。利用Chen， Lu和Pope的局部解构造了一个显式非奇异完全环形Calabi-Yau度量。这个度量给出了一个新的超引力解，表示D3膜。我们构造了一个新的无限颤振规范理论族，它可以追溯到最近由Hannany， Kazakopoulos和wect发现的颤振规范理论。这个系列包括第三个del Pezzo曲面的颤振规范理论。我们已经用体积最小化证明了，这些理论一般都有不合理的r电荷。AdS/CFT对应意味着对偶几何是不规则的环形Sasaki-Einstein流形。利用逆散射方法研究了五维真空爱因斯坦方程的稳态和轴对称孤子解。我们以五维闵可夫斯基时空为种子，再现了三岛和井口最近发现的带有旋转双球的黑环解。

We aimed to study compact Einstein manifolds and Riemannian manifolds of special holonomy from a view point of black hole.These geometries are physically interpreted as higher dimensional gravitational instantons, and deeply related to new geometries expected by string theory. Especially, compact Sasaki-Einstein geometry has been extensively studied in connection with AdS/CFT correspondence.1. We have constructed a new infinite series of Einstein metrics on sphere bundles. These metrics appear as a certain limit of 5-dimensional AdS Kerr black holes. In the special case, the metric gives a homogeneous Sasaki-Einstein metric. Our construction is a higher dimensional version of the method of Page, which gave an inhomogeneous Einstein metric on the first del Pezzo surface.2. Recently, D3 branes at the tip of toric Calabi-Yau cones have been extensively studied from a view point of AdS/CFT correspondence. It is natural to consider the deformations of cone metrics in order to explore non-conformal field theories. We have constructed an explicit non-singular complete toric Calabi-Yau metric using the local solution found by Chen, Lu and Pope. This metric gives a new supergravity solution representing D3 branes.3. We have constructed a new infinite family of quiver gauge theories which blow down to the quiver gauge theories found by Hannany, Kazakopoulos and Wecht, recently. This family includes a quiver gauge theory for the third del Pezzo surface. We have shown, using volume minimaization, that these theories generically have irrational R-charges. The AdS/CFT correspondence implies that the dual geometries are irregular toric Sasaki-Einstein manifolds.4. We studied stationary and axially symmetric solitonic solutions of five dimensional vacuum Einstein equations by using the inverse scattering method. From five dimensional Minkowski space-time as a seed, we reproduced a black ring solution with a rotating two sphere which was found by Mishima and Iguchi recently.