Ricardo Troncoso
In collaboration with
Hernán González and David Tempo
arXiv:1107.3647 [hep-th]

Field theories with anisotropic scaling in 2d
D :
P :
H :
Two-dimensional Lifshitz algebra with dynamical exponent z :

Key observation Isomorphism :

Field theories with anisotropic scaling in 2d
On a cylinder :
Finite temperature (torus) :
Change of basis : swaps the roles of Euclidean time and the angle
Does not fit the cylinder (yet !)

On a cylinder :
Finite temperature (torus) :

Field theories with anisotropic scaling in 2d (finite temperature)
High-Low temperature duality :
Relationship for Z at low and high temperatures :
Hereafter we will then assume that
Note that for z=1 reduces to the well known
S-modular invariance for chiral movers in CFT !

Asymptotic growth of the number of states
• Let’s assume a gap in the spectrum
•
Ground state energy is also assumed to be negative :
Therefore, at low temperatures :
Generalized S-mod. Inv. :
At high temperatures :

High T
•
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 :

Asymptotic growth of the number of states
Note that for z=1 reduces to Cardy formula *
* Shifted Virasoro operator
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 states

Anisotropic holography
Lifshitz spacetime in 2+1 (KLM):
Characterized by l , z . Reduces to AdS for z = 1
Isometry group:

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

Key observation + High-Low Temp. duality (Holographic version)
Coordinate transformation :
Both are diffeomorphic provided :

Anisotropic holography:
Solitons and the microscopic entropy of 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 of asymptotically Lifshitz black holes
Euclidean action
(Soliton) :
Euclidean action
(black hole) :

Euclidean action
(black hole) :

Black hole entropy :
Perfect matching provided :
Field theory entropy:

An explicit example : BHT Massive Gravity
E. A. Bergshoeff, O. Hohm, P. K. Townsend, PRL 2009
Let’s focus on the special case :
The theory admits Lifshitz spacetimes with

An explicit example : BHT Massive Gravity
Special case :
Asymptotically Lifshitz black hole :
E. Ayón-Beato, A. Garbarz, G. Giribet and M. Hassaine, PRD 2009

An explicit example : BHT Massive Gravity
Special case :
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 action
O. Hohm and E. Tonii, JHEP 2010
Regularization intended for the black hole with z = 3, l
It must necessarily work for the soliton ! (z = 1/3, l/3)

Regularized Euclidean action
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 mass :
Black hole entropy :

Black hole entropy (microcanonical ensemble)
Perfect matching with field theory entropy
(z = 3) provided

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

Remarks :
•
Black hole and soliton metrics do not match at infi
•
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 soliton and vice versa

Asymptotic growth of the number of states
•
Canonical ensemble, 1 st law :
Reduces to Stefan-Boltzmann for z=1