Anisotropic holography Anisotropic holography

and the microscopic entropyand the microscopic entropy

of Lifshitz black holes in 3D of Lifshitz black holes in 3D

Ricardo TroncosoRicardo Troncoso

In collaboration withIn collaboration with

Hernán González and David TempoHernán González and David Tempo

Centro de Estudios Científicos (CECS) Centro de Estudios Científicos (CECS) Valdivia, Valdivia, Chile Chile arXiv:1107.3647 [hep-arXiv:1107.3647 [hep-th]th]

Field theories with anisotropic scaling Field theories with anisotropic scaling in 2d in 2d

Two-dimensional Lifshitz algebra with dynamical exponent z :

D :

P :

H :

Isomorphism : Isomorphism :

This isomorphism induces the equivalence of Zbetween low and high T

Key observation

Field theories with anisotropic scaling Field theories with anisotropic scaling in 2d in 2d

Change of basis :

Finite temperature (torus) Finite temperature (torus) : :

On a cylinder On a cylinder : :

swaps the roles of Euclidean time and the angle

Does not fit the cylinder (yet !)

Finite temperature (torus) Finite temperature (torus) : :

On a cylinder On a cylinder : :

Field theories with anisotropic scaling in 2d (finite Field theories with anisotropic scaling in 2d (finite temperature) temperature)

Relationship for Z at low and high temperatures :

Hereafter we will then assume that

High-Low temperature duality :

Note that for z=1 reduces to the well known S-modular invariance for chiral movers in CFT !

Therefore, at low temperatures :

• Let’s assume a gap in the spectrum

• Ground state energy is also assumed to be negative :

Generalized S-mod. Inv. :

At high temperatures :

Asymptotic growth of the number of Asymptotic growth of the number of statesstates

• Asymptotic growth of the number of states at fixed energy is then obtained from :

The desired result is easily obtained in the saddle point approximation :

High T

* Shifted Virasoro operatorCardy formula is expressed only through its …fixed and lowest eigenvalues. The N° of states can be obtained from the spectrum without making any explicit reference to the central charges !

Asymptotic growth of the number of Asymptotic growth of the number of statesstates

Note that for z=1 reduces to Cardy formula *

Asymptotic growth of the number of Asymptotic growth of the number of statesstates

• Remarkably, asymptotically Lifshitz black holes in 3D precisely fit these results !

• The ground state is a gravitational soliton

Lifshitz spacetime in 2+1 (KLM):

Characterized by l , z . Reduces to AdS for z = 1

Isometry group:

Anisotropic Anisotropic holographyholography

Anisotropic Anisotropic holographyholographyKey observation + High-Low Temp. duality

(Holographic version)

Key observation + High-Low Temp. duality (Holographic version)

Coordinate transformation :

Both are diffeomorphic provided :

Anisotropic holography: Anisotropic holography:

Solitons and the microscopic entropy Solitons and the microscopic entropy

of asymptotically Lifshitz black holesof asymptotically Lifshitz black holes

• The previous procedure is purely geometrical : Result remains valid regardless the theory !

• Asymptotically (Euclidean) Lifshitz black holes in 2+1 become diffeomorphic to gravitational solitons with :

Lorentzian soliton : Regular everywhere.no CTCs once is unwrapped.

Fixed mass (integration constant reabsorbed by rescaling).

It becomes then natural to regard the soliton as the corresponding ground state.

Solitons and the microscopic entropy Solitons and the microscopic entropy

of asymptotically Lifshitz black holesof asymptotically Lifshitz black holes

Euclidean action (Soliton) :

Euclidean action (black hole) :

Euclidean action (black hole) :

Black hole entropy :

Field theory entropy:

Perfect matching provided :

Let’s focus on the special case :

E. A. Bergshoeff, O. Hohm, P. K. Townsend, PRL 2009

An explicit example : BHT Massive An explicit example : BHT Massive GravityGravity

The theory admits Lifshitz spacetimes with

Special case :

An explicit example : BHT Massive An explicit example : BHT Massive GravityGravity

Asymptotically Lifshitz black hole :E. Ayón-Beato, A. Garbarz, G. Giribet and M. Hassaine, PRD 2009

Special case :

An explicit example : BHT Massive An explicit example : BHT Massive GravityGravity

Asymptotically Lifshitz gravitational soliton :

• Regular everywhere:

• Geodesically complete.

• Same causal structure than AdS

• Asymptotically Lifshitz spacetime with :

• Devoid of divergent tidal forces at the origin !

Euclidean asymptotically Lifshitz black hole is diffeomorphic to the gravitational soliton :

Coordinate transformation :

Followed by :

Regularized Euclidean Regularized Euclidean actionaction

Regularization intended for the black hole with z = 3, lIt must necessarily work for the soliton ! (z = 1/3, l/3)

O. Hohm and E. Tonii, JHEP 2010

Regularized Euclidean Regularized Euclidean actionaction

Gravitational soliton :

Finite action :

Fixed mass :

Black hole : (Can be obtained from the soliton + High Low Temp. duality)

Finite action :

Black hole mass :

Black hole entropy :

Black hole mass :

Perfect matching with field theory entropy

(z = 3) provided

Black hole entropy (microcanonical Black hole entropy (microcanonical ensemble)ensemble)

• Ending remarks: Specific heat, “phase transitions” and an extension of cosmic censorship.

• Black hole and soliton metrics do not match at infi…nity

• An obstacle to compare them in the same footing ?

• True for generically different z, l .

• Remarkably, for

circumvented since their Euclidean versions are diffeomorphic.

• The moral is that, any suitably regularized Euclidean action for the black hole is necessarily …finite for the gravitational soliton and vice versa

Remarks Remarks ::

Asymptotic growth of the number of Asymptotic growth of the number of statesstates

Reduces to Stefan-Boltzmann for z=1

• Canonical ensemble, 1st law :