Volumes of Compact Manifolds

TL;DR

This paper systematically computes the volumes of various compact manifolds relevant in physics, such as spheres, projective spaces, and flag manifolds, providing normalized scales useful for multiple physical applications.

Contribution

It introduces a standardized approach to calculating manifold volumes with natural normalizations, applicable to spheres, products, and quotients, aiding physical theories and models.

Findings

Explicit volume formulas for key manifolds

Normalized scales for physical applications

Useful for instanton and string theory calculations

Abstract

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.