Warped flatland
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Springer
Received: April 28, Revised: September 17, Accepted: October 4, Published: November 13,
2020 2020 2020 2020
St´ ephane Detournay,a Wout Merbis,a Gim Seng Ngb,c and Raphaela Wutted a
Physique Math´ematique des Interactions Fondamentales, Universit´e Libre de Bruxelles, and International Solvay Institutes, Campus Plaine – CP 231, 1050 Bruxelles, Belgium b School of Mathematics, Trinity College Dublin, Dublin 2, Ireland c Hamilton Mathematical Institute, Trinity College Dublin, Dublin 2, Ireland d Institute for Theoretical Physics, TU Wien, Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We study warped flat geometries in three-dimensional topologically massive gravity. They are quotients of global warped flat spacetime, whose isometries are given by the 2-dimensional centrally extended Poincar´e algebra. The latter can be obtained as a certain scaling limit of Warped AdS3 space with a positive cosmological constant. We discuss the causal structure of the resulting spacetimes using projection diagrams. We study their charges and thermodynamics, together with asymptotic Killing vectors preserving a consistent set of boundary conditions including them. The asymptotic symmetry group is given by a Warped CFT algebra, with a vanishing current level. A generalization of the derivation of the Warped CFT Cardy formula applies in this case, reproducing the entropy of the warped flat cosmological spacetimes. Keywords: Black Holes, Gauge-gravity correspondence, Space-Time Symmetries ArXiv ePrint: 2001.00020
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)061
JHEP11(2020)061
Warped flatland
Contents 1 Introduction
1 4 5 6 8 8 8 12
3 Warped flat spacetimes in TMG 3.1 First-order formulation of TMG 3.2 Phase space 3.3 Asymptotic symmetry transformations 3.4 Charge algebra 3.5 Thermodynamics
14 14 15 16 18 18
4 Warped conformal field theories at vanishing level
20
5 Discussion
22
A Limits from WAdS or WdS to warped flat spacetimes A.1 Limits from WAdS or WdS A.2 Limit from WAdS black holes with positive cosmological constant A.3 Warped flat limit in the quadratic ensemble
24 24 24 26
B Asymptotic symmetries of WAdS3
28
C Derivations of warped Cardy entropy formula
30
1
Introduction
The identification of the microscopic origin of gravitational entropy is one of the most fertile puzzles in modern theoretical physics. Progress in this context has been especially significant when it involves the identification of black hole microstates, however, the same issue for cosmological horizons has remained more elusive so far. In the former case, an important breakthrough appeared in the seminal derivation of the microscopic entropy of five-dimensional extremal BPS black holes in the context of string theory by Strominger and Vafa [1]. This result has pointed at the central role played by two-dimensional conformal
–1–
JHEP11(2020)061
2 Geometry of warped flat s
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