Renormalized Volume

Abstract

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.