Geometry of Higher Dimensional Black Hole and Compact Einstein Space

我们的目标是从黑洞的角度研究紧致的Einstein流形和特殊完整性的Riemann流形，这些几何在物理上被解释为高维引力瞬子，并且与弦理论所期望的新几何有着深刻的联系。特别是，紧凑的Sasaki-Einstein几何已被广泛研究与AdS/CFT对应。我们构造了一个新的球丛上的Einstein度规无穷级数。这些度规表现为5维AdS Kerr黑洞的某种极限。在特殊情况下，该度规给出了齐次的Sasaki-Einstein度规。我们的构造是佩奇方法的更高维度版本，该方法在第一个del Pezzo表面上给出了非齐次爱因斯坦度规。2.最近，从AdS/CFT对应的观点出发，人们对复曲面Calabi-Yau锥顶端的D3膜进行了广泛的研究。为了探索非共形场论，考虑锥度规的变形是很自然的。利用Chen，Lu和Pope给出的局部解，我们构造了一个显式的非奇异完全环面Calabi-Yau度量.这个度规给出了一个新的代表D3膜的超引力解。我们构造了一个新的无限族的超规范理论，它与Hannany，Kazakopoulos和Wecht最近发现的超规范理论是一致的。这个族包括第三del Pezzo曲面的规范理论。我们已经表明，使用体积极小化，这些理论一般有无理R-收费。AdS/CFT对应意味着对偶几何是不规则复曲面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定态和轴对称孤子解。以五维Minkowski时空为种子，再现了三岛和井口最近发现的双球旋转的黑环解.

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.