重整体积
Renormalized Volume
摘要：
我们开发了一个通用的分布演算的规管体积的度量，沿超曲面是奇异的。当超曲面是一个共形无穷远时，我们给出了简单的积分分布表达式的发散和异常的调节体积功能有效的任何选择的调节器。对于闭超曲面或共形紧致几何，以前发展的共形紧致流形的边界演算方法可以应用于给出表示为局部量上的超曲面积分的发散和反常的显式全息公式（该方法也扩展到非闭超曲面）。由此产生的异常不依赖于任何特定的选择的监管机构，而监管机构的依赖性的分歧，正是由这些公式。共形超曲面不变量的研究可以通过要求奇异度量服从，光滑和形式上的一个合适的顺序，Yamabe型问题的边界数据沿共形无穷远。我们证明了这些奇异Yamabe解决方案的体积异常是一个共形不变的积分的局部Q-曲率，广义的布兰森Q-曲率，包括数据的嵌入。在每一个维度上，这都规范地定义了Willmore能量/刚性弦作用的高维推广。最近格雷厄姆证明，第一变化的体积异常恢复密度阻碍光滑的解决方案，这个奇异的Yamabe问题，我们给出了一个新的证明，这一结果采用我们的边界演算。我们的结果的物理应用包括纠缠熵的量子修正的研究。
英文摘要：
We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.
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