Renormalized volumes with boundary

Abstract

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré-Einstein structure, this result recovers Branson's Q curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

Einstein gravity from Conformal Gravity in 6D

“…In addition to providing an explicit realization of the CG action and being used in the definition of renormalized volume, conformal invariants are considered in the computation of the type-B anomalies of a CFT and play a prominent role in the context of conformal geometry, as well [9,10,41,[45][46][47][48][49][50][51].…”

Einstein-AdS action, renormalized volume/area and holographic Rényi entropies

“…In the context of asymptotically hyperbolic (AH) manifolds [4][5][6][7][8][9], its resolution is known to provide relevant information about the boundary geometry, in the form of conformal invariant quantities. More recently and by means of tractor calculus techniques [10,11], the renormalized volume problem has been extended to more general setups in [12,13], where hidden algebraic structures are enhanced by considering the bulk itself as a conformal manifold.…”

Renormalized entanglement entropy and curvature invariants

“…The latter topics has been explored right from the very early days of the AdS/CFT correspondence [15,16], as it is also relevant to the holographic computation of Wilson loops. Within the mathematics community, there has been considerable study of renormalized areas of surfaces, see for example [17][18][19][20][21][22]. Connections between renormalized areas, entanglement and the Willmore functional have been explored within both the mathematics and the physics communities [23,24].…”

“…Contraction of the Gauss-Codazzi relations gives 19) where here and in the rest of this sectoin we show the normal index s as a subscript to improve the clarity of equations. Contracting further gives…”