Asymptotic boundary conditions and square integrability in the partition function of AdS gravity
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Received: April 14, 2020 Accepted: June 10, 2020 Published: June 29, 2020
Asymptotic boundary conditions and square integrability in the partition function of AdS gravity
a
Departamento de Matem´ atica-FCEN-UBA & IMAS-CONICET, Ciudad Universitaria, pabell´ on 1, 1428, Buenos Aires, Argentina b Departamento de F´ısica-FCEN-UBA & IFIBA-CONICET, Ciudad Universitaria, pabell´ on 1, 1428, Buenos Aires, Argentina c Instituto de Astronom´ıa y F´ısica del Espacio (IAFE), Pabell´ on IAFE-CONICET, Ciudad Universitaria, C.C. 67 Suc. 28, Buenos Aires, Argentina
E-mail: [email protected], [email protected], [email protected], [email protected] 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. Keywords: BRST Quantization, Field Theories in Lower Dimensions, Models of Quantum Gravity ArXiv ePrint: 2004.01723
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)172
JHEP06(2020)172
Joel Acosta,a Alan Garbarz,b Andr´ es Goyab and Mauricio Lestonc
Contents 1
2 Metric perturbations
3
3 Ghosts and asymptotic vector fields
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4 Square integrability in higher dimensions
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5 Summary
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1
Introduction
Three dimensional pure gravity with negative cosmological constant has been widely used as a toy model for addressing problems of quantum gravity (see for example [1–4]). One important step towards the understanding of the quantum description of AdS 3 gravity was taken in [3], where it was com
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