标题： 紧致流形的体积 显示英文标题

标题： Volumes of Compact Manifolds

摘要： 我们系统地计算了在物理学中出现的紧致流形的体积：球面、射影空间、群流形和广义旗流形。 在每种情况下，我们说明了我们认为最自然的流形尺度或归一化方式，即球面单位半径条件的推广。 为此，我们首先用一些参数描述流形，建立一个度量，该度量诱导出体积元素，并对参数的适当范围进行积分；在大多数情况下，我们的流形将是球面或（扭曲）球面的乘积，或者是球面的商（齐性空间）。 我们的结果在多个物理实例中应该很有用，如瞬子计算、弯曲空间中的传播子、σ模型、齐性流形中的几何散射、纠缠态的密度矩阵等。 一些旗流形最近也作为例外全纯性流形出现；紧致爱因斯坦流形的体积出现在弦理论中。 显示英文摘要

摘要： We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius condition for spheres. For this aim we first describe the manifold with some parameters, set up a metric, which induces a volume element, and perform the integration for the adequate range of the parameters; in most cases our manifolds will be either spheres or (twisted) products of spheres, or quotients of spheres (homogeneous spaces). Our results should be useful in several physical instances, as instanton calculations, propagators in curved spaces, sigma models, geometric scattering in homogeneous manifolds, density matrices for entangled states, etc. Some flag manifolds have also appeared recently as exceptional holonomy manifolds; the volumes of compact Einstein manifolds appear in String theory.