Symmetries at null boundaries: two and three dimensional gravity cases
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Springer
Received: August 2, Revised: September 5, Accepted: September 14, Published: October 16,
2020 2020 2020 2020
H. Adami,a M.M. Sheikh-Jabbari,a V. Taghiloo,b H. Yavartanooc and C. Zwikeld a
School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran b Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45137-66731, Zanjan, Iran c Beijing Institute of Mathematical Sciences and Applications (BIMSA), Huairou District, Beijing 101408, P.R. China d Institute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We carry out in full generality and without fixing specific boundary conditions, the symmetry and charge analysis near a generic null surface for two and three dimensional (2d and 3d) gravity theories. In 2d and 3d there are respectively two and three charges which are generic functions over the codimension one null surface. The integrability of charges and their algebra depend on the state-dependence of symmetry generators which is a priori not specified. We establish the existence of infinitely many choices that render the surface charges integrable. We show that there is a choice, the “fundamental basis”, where the null boundary symmetry algebra is the Heisenberg⊕Diff(d − 2) algebra. We expect this result to be true for d > 3 when there is no Bondi news through the null surface. Keywords: AdS-CFT Correspondence, Black Holes in String Theory, Classical Theories of Gravity, Gauge-gravity correspondence ArXiv ePrint: 2007.12759
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)107
JHEP10(2020)107
Symmetries at null boundaries: two and three dimensional gravity cases
Contents 1 Introduction
1
2 Null boundary symmetry (NBS) algebra, 2d dilaton gravity case 2.1 NBS generating vector fields 2.2 Surface charges 2.3 Integrable basis for the charges
4 5 6 7 8 10 11 12
4 Change of basis for integrable charges 4.1 Change of basis, general formulation 4.2 2d gravity — generic case 4.3 3d gravity — general discussion and fundamental basis for NBS algebra 4.3.1 3d NBS algebra in “fundamental basis” 4.3.2 Further integrable Lie-algebras
15 16 18 19 20 20
5 Discussion and outlook
23
A Classification of solutions to 2d dilaton gravity
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B Classification of AdS3 solutions
29
C Modified bracket and integrable parts of charges
31
1
Introduction
A standard way to describe a d-dimensional field theory at the classical level is through an action which is the integral of a Lagrangian over a d-dimensional manifold M. For physical theories, M is typically a Lorentzian manifold which may have a codimension one boundary ∂M. The variational principle can then be used to derive equations of motion which are typically second order differential equations over M. These equations are wellposed if their solutions can be full
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