Asymptotic boundary conditions and square integrability in the partition function of AdS gravity

Joel Acosta,Alan Garbarz,Mauricio Leston,Andrés Goya

doi:10.1007/jhep06(2020)172

Abstract

There has been renewed interest in the path-integral computation of the partition function of AdS3 gravity, both in the metric and Chern-Simons formulations. The one-loop partition function around Euclidean AdS3 turns out to be given by the vacuum character of Virasoro group. This stems from the work of Brown and Henneaux (BH) who showed that, in AdS3 gravity with sensible asymptotic boundary conditions, an infinite group of (improper) diffeomorphisms arises which acts canonically on phase space as two independent Virasoro symmetries. The gauge group turns out to be composed of so-called “proper” diffeomorphisms which approach the identity at infinity fast enough. However, it is sometimes far from evident to identify where BH boundary conditions enter in the path integral, and much more difficult to see how the improper diffeomorphisms are left out of the gauge group. In particular, in the metric formulation, Giombi, Maloney and Yin obtained the one-loop partition function around thermal AdS3 resorting to the heat kernel method to compute the determinants coming from the path integral. Here we identify how BH boundary conditions follow naturally from the usual requirement of square-integrability of the metric perturbations. Also, and equally relevant, we clarify how the quotient by only proper diffeomorphisms is implemented, promoting the improper diffeomorphisms to symmetries in the path integral. Our strategy is general enough to apply to other approaches where square integrability is assumed. Finally, we show that square integrability implies that the asymptotic symmetries in higher dimensional AdS gravity are just isometries.

Highlights

JHEP06(2020)172 asymptotic symmetries of AdS are nowadays understood, in the context of AdS/CFT [9], as the local conformal group of the boundary theory
This stems from the work of Brown and Henneaux (BH) who showed that, in AdS3 gravity with sensible asymptotic boundary conditions, an infinite group of diffeomorphisms arises which acts canonically on phase space as two independent Virasoro symmetries
It is sometimes far from evident to identify where BH boundary conditions enter in the path integral, and much more difficult to see how the improper diffeomorphisms are left out of the gauge group

Summary

Metric perturbations

We are interested in understanding the phase space of metric perturbations that is relevant for the tion of cgoramvpituytaistioronuogfhtlyheoof nteh-elofooprmpartDitigoen−fund3cxt√iogn(Ro+f2A).dTS3hegrianvteitgyr.aTndheinpathrteiteioxnpofnuennctshould be expanded around the hyperbolic metric up to second order and this gives an. The functional integral should sum over all those h that satisfy a sensible boundary condition and belong to some judicious gauge choice. We will scrutinize these requirements in this and the following section. These are the usual Brown-Henneaux asymptotic vector fields, with the exception of a milder fall-off condition O(z3) instead of O(z4) in the non-radial components. We should emphasize that in order to obtain the Virasoro symmetry what it really matters is how the algebra of vectors is represented on the space of metrics, and not the precise form of the vectors or the metric components. This indicates that the vectors act with the coadjoint representation of the Virasoro algebra

Ghosts and asymptotic vector fields

Square integrability in higher dimensions

Summary