Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies

Ojeda, Emilio; Pérez, Alfredo

doi:10.1007/jhep08(2019)079

Cited by 30 publications

(30 citation statements)

References 65 publications

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“…While we considered the specific example of two asymptotically anti-de Sitter regions, most of our considerations on holonomies and radial Wilson lines qualitatively apply whenever there are two boundaries, independently of the form that the boundary conditions explicitly take there. So, in particular, one boundary can describe an asymptotically anti-de Sitter region and the other can describe a black hole horizon, along the lines of [56,57] (see also [58,59]). Although we have not explored the problem, similar features are also expected to hold with more boundaries.…”

Section: Discussionmentioning

confidence: 99%

Asymptotic dynamics of AdS3 gravity with two asymptotic regions

Henneaux

Merbis

Ranjbar

2020

J. High Energ. Phys.

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The asymptotic dynamics of AdS 3 gravity with two asymptotically anti-de Sitter regions is investigated, paying due attention to the zero modes, i.e., holonomies along non-contractible circles and their canonically conjugates. This situation covers the eternal black hole solution. We derive how the holonomies around the non-contractible circles couple the fields on the two different boundaries and show that their canonically conjugate variables, needed for a consistent dynamical description of the holonomies, can be related to Wilson lines joining the boundaries. The action reduces to the sum of four free chiral actions, one for each boundary and each chirality, with additional non-trivial couplings to the zero modes which are explicitly written. While the Gauss decomposition of the SL(2, R) group elements is useful in order to treat hyperbolic holonomies, the Iwasawa decomposition turns out to be more convenient in order to deal with elliptic and parabolic holonomies. The connection with the geometric action is also made explicit. Although our paper deals with the specific example of two asymptotically anti-de Sitter regions, most of our global considerations on holonomies and radial Wilson lines qualitatively apply whenever there are multiple boundaries, independently of the form that the boundary conditions explicitly take there.An interesting feature of the analysis is that the holonomy is dynamical and possesses a conjugate momentum. This conjugate momentum can be related to the radial Wilson lines connecting the boundaries. The phase space of the system is therefore not just two copies of the phase spaces of the boundary theories, but there is in addition the global dynamical zero modes described by the holonomy and the radial Wilson lines. In the quantum theory, the Hilbert space does not factorize into the mere tensor product of the boundary Hilbert spaces, but involves also the zero modes. The non-trivial link between the boundary Hilbert spaces does not need to be implemented by hand since it follows from the action, which is not given by mere multiple copies of one boundary action, but possesses an extra piece coupling the dynamical global zero modes to the boundary fields.It turns out that the resulting reduced boundary theory can be viewed as a dynamical theory of the boundary Virasoro charges, which, together with the zero modes, completely capture the physics of the system. These charges transform in the coadjoint representation of the Virasoro group and the boundary dynamics takes place on the coadjoint orbits.This leads to two different descriptions of the boundary dynamics. One is in terms of chiral bosons, which can be viewed as providing Darboux-like coordinates on the coadjoint orbits. Since there are two SL(2, R) factors and two boundaries, one ends up with four chiral bosons theories, one per SL(2, R) factor and boundary. We show that the chiral bosons corresponding to the same SL(2, R) at two different boundaries are linked by the fact that the conjugate momentum to the holonomy (which is ...

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Section: Discussionmentioning

confidence: 99%

Asymptotic dynamics of AdS3 gravity with two asymptotic regions

Henneaux

Merbis

Ranjbar

2020

J. High Energ. Phys.

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“…Different sets of boundary conditions whose chemical potentials also depend on the dynamical fields have been devised in order to make contact with two-dimensional integrable systems in refs. [25][26][27][28][29].…”

Section: Jhep11(2020)015mentioning

confidence: 99%

Gravitational dual of averaged free CFT’s over the Narain lattice

Pérez

Troncoso

2020

J. High Energ. Phys.

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It has been recently argued that the averaging of free CFT’s over the Narain lattice can be holographically described through a Chern-Simons theory for U (1)D×U (1)D with a precise prescription to sum over three-dimensional handlebodies. We show that a gravitational dual of these averaged CFT’s would be provided by Einstein gravity on AdS3 with U (1)D−1× U (1)D−1 gauge fields, endowed with a precise set of boundary conditions closely related to the “soft hairy” ones. Gravitational excitations then go along diagonal SL (2, ℝ) generators, so that the asymptotic symmetries are spanned by U (1)D× U (1)D currents. The stress-energy tensor can then be geometrically seen as composite of these currents through a twisted Sugawara construction. Our boundary conditions are such that for the reduced phase space, there is a one-to-one map between the configurations in the gravitational and the purely abelian theories. The partition function in the bulk could then also be performed either from a non-abelian Chern-Simons theory for two copies of SL (2, ℝ) × U (1)D−1 generators, or formally through a path integral along the family of allowed configurations for the metric. The new boundary conditions naturally accommodate BTZ black holes, and the microscopic number of states then appears to be manifestly positive and suitably accounted for from the partition function in the bulk. The inclusion of higher spin currents through an extended twisted Sugawara construction in the context of higher spin gravity is also briefly addressed.

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“…Other examples of this relationship between 2D integrable systems and gravity in 2+1, have been also made for the cases of "flat" and "soft hairy" boundary conditions in[25] and[26], respectively.…”

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confidence: 99%

BTZ black hole with Korteweg–de Vries-type boundary conditions: Thermodynamics revisited

2019

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The thermodynamic properties of the Bañados-Teitelboim-Zanelli (BTZ) black hole endowed with Korteweg-de Vries (KdV)-type boundary conditions are considered. This familiy of boundary conditions for General Relativity on AdS 3 is labeled by a nonnegative integer n, and gives rise to a dual theory which possesses anisotropic Lifshitz scaling invariance with dynamical exponent z = 2n + 1. We show that from the scale invariance of the action for stationary and circularly symmetric spacetimes, an anisotropic version of the Smarr relation arises, and we prove that it is totally consistent with the previously reported anisotropic Cardy formula. The set of KdV-type boundary conditions defines an unconventional thermodynamic ensemble, which leads to a generalized description of the thermal stability of the system. Finally, we show that at the self-dual temperature T s = 1 2π ( 1 z ) z z+1 , there is a Hawking-Page phase transition between the BTZ black hole and thermal AdS 3 spacetime. arXiv:1907.13026v2 [hep-th]

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Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies

Cited by 30 publications

References 65 publications

Asymptotic dynamics of AdS3 gravity with two asymptotic regions

Asymptotic dynamics of AdS3 gravity with two asymptotic regions

Gravitational dual of averaged free CFT’s over the Narain lattice

BTZ black hole with Korteweg–de Vries-type boundary conditions: Thermodynamics revisited

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