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)