Two Virasoro symmetries in stringy warped AdS3

Journal of High Energy Physics, Dec 2014

Abstract We study three-dimensional consistent truncations of type IIB supergravity which admit warped AdS3 solutions. These theories contain subsectors that have no bulk dynamics. We show that the symplectic form for these theories, when restricted to the non-dynamical subsectors, equals the symplectic form for pure Einstein gravity in AdS3. Consequently, for each consistent choice of boundary conditions in AdS3, we can define a consistent phase space in warped AdS3 with identical conserved charges. This way, we easily obtain a Virasoro × Virasoro asymptotic symmetry algebra in warped AdS3; two different types of Virasoro × Kač-Moody symmetries are also consistent alternatives. Next, we study the phase space of these theories when propagating modes are included. We show that, as long as one can define a conserved symplectic form without introducing instabilities, the Virasoro × Virasoro asymptotic symmetries can be extended to the entire (linearised) phase space. This implies that, at least at semi-classical level, consistent theories of gravity in warped AdS3 are described by a two-dimensional conformal field theory, as long as stability is not an issue.

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Two Virasoro symmetries in stringy warped AdS3

Geoffrey Comp`ere 0 1 3 6 Monica Guica 0 1 3 4 Maria J. Rodriguez 0 1 2 3 5 Philadelphia 0 1 3 PA 0 1 3 U.S.A. 0 1 3 0 Open Access , c The Authors 1 Cambridge , MA 02138 , U.S.A 2 Institut de Physique Th eorique, CEA Saclay 3 Universit e Libre de Bruxelles and International Solvay Institutes 4 David Rittenhouse Laboratory, University of Pennsylvania 5 Center for the Fundamental Laws of Nature, Harvard University 6 Physique Th eorique et Math ematique We study three-dimensional consistent truncations of type IIB supergravity which admit warped AdS3 solutions. These theories contain subsectors that have no bulk dynamics. We show that the symplectic form for these theories, when restricted to the non-dynamical subsectors, equals the symplectic form for pure Einstein gravity in AdS3. Consequently, for each consistent choice of boundary conditions in AdS3, we can define a consistent phase space in warped AdS3 with identical conserved charges. This way, we easily obtain a Virasoro Virasoro asymptotic symmetry algebra in warped AdS3; two different types of Virasoro Kac-Moody symmetries are also consistent alternatives. 1 Introduction 2 Setup and review 3 Phase spaces without bulk propagating modes The S-dual dipole truncation The NHEMP truncation Review of the covariant phase space formalism Explicit expressions for the symplectic structure and charges Universal solution space without bulk propagating modes Equivalence of the warped and unwarped symplectic structures Dirichlet boundary conditions Dirichlet-Neumann chiral boundary conditions 4 Including the bulk propagating modes Behaviour of the linearized bulk solutions Phase space for the S-dual dipole theory 5 A Details of the computations A.1 The linearized solution for boundary gravitons in radial gauge A.2 Linear perturbations in the NHEMP truncation Black holes carry an entropy given by a remarkably simple, yet completely universal, formula: the area of their event horizon in Planck units, divided by four [1, 2]. Despite many years of investigation, the microscopic origin of this formula and the reason behind its universality have only been understood for a very special class of black holes, namely those whose near-horizon region contains an AdS3 factor [3, 4]. The microstates of such black holes correspond to thermal excitations in a two-dimensional conformal field theory (CFT2) and their entropy is counted by Cardys formula [5], a universal CFT2 formula that perfectly matches the area law on the gravity side. Moreover, the symmetries of the microscopic CFT two copies of the infinite-dimensional Virasoro algebra are directly visible in spacetime in the form of asymptotic symmetries [6]. Unfortunately, no black hole in nature has an AdS3 factor in its near-horizon region. Nevertheless, a few years ago it was proposed that extremal Kerr black holes astrophysical examples of which do seem to exist1 are described by a (chiral half of a) two-dimensional CFT [12]. This proposal was based on the study of the nearhorizon scaling limit of the extremal Kerr black hole [13] (henceforth abbreviated as NHEK), which enjoys an enlarged symmetry group, namely SL(2, R)L U(1)R. The main supporting evidence was the enhancement of the U(1)R isometry to the full Virasoro algebra at the level of asymptotic symmetries and the perfect match between the Bekenstein-Hawking entropy of the black hole and the Cardy entropy of the putative dual CFT. These results were soon extended (see the reviews [14, 15]) to very general extremal black holes, indicating the existence of a universal holographic correspondence for such black holes. The part of the geometry that appears to play the key role in the duality is a warped AdS3 factor whose structure is that of a U(1) fibre over AdS2 which is universally present in the near-horizon region of extremal black holes [16]. Despite the remarkable agreement between the microscopic and macroscopic entropy of extremal black holes, several puzzles remain, regarding the very applicability of Cardys formula to the microscopic counting. First, Cardys formula applies to two-sided CFT2s, whose symmetries consist of both a left-moving and a right-moving Virasoro algebra. Nevertheless, all attempts to find an asymptotic symmetry group for NHEK (or, more generally, for warped AdS3) that contains both copies of the Virasoro algebra simultaneously have so far failed.2 Instead, boundary conditions for warped AdS3 spacetimes have been found [23, 24], which admit as asymptotic symmetries a left-moving Virasoro algebra, together with a U(1) Kac-Moody algebra that enhances the right-moving translations. These results have inspired the search for two-dimensional QFTs sometimes denoted as warped CFTs that exhibit these symmetries. While no consistent quantum example of a warped CFT is known to date (see, however, the semi-classical examples in [25]), on very high temperatures is dictated by a universal, Cardy-like formula.3 general grounds, [26] (...truncated)


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Geoffrey Compère, Monica Guica, Maria J. Rodriguez. Two Virasoro symmetries in stringy warped AdS3, Journal of High Energy Physics, 2014, pp. 12, Volume 2014, Issue 12, DOI: 10.1007/JHEP12(2014)012