Anisotropic holography

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Anisotropic holography and the microscopic
entropy
of Lifshitz black holes in 3D
Ricardo Troncoso In collaboration with Hernán
González and David Tempo
Centro de Estudios Científicos (CECS)
Valdivia, Chile
arXiv1107.3647 hep-th
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Field theories with anisotropic scaling in 2d
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Isomorphism
Key observation
This isomorphism induces the equivalence of
Z between low and high T
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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 !)
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On a cylinder
Finite temperature (torus)
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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 z1 reduces to the well known
S-modular invariance for chiral movers in CFT !
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Asymptotic growth of the number of states

• Lets 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
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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
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Asymptotic growth of the number of states
Note that for z1 reduces to Cardy formula
Shifted Virasoro operator Cardy 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 !
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Asymptotic growth of the number of states

• Remarkably, asymptotically Lifshitz black holes
  in 3D
• precisely fit these results !
• The ground state is a gravitational soliton

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Anisotropic holography
Lifshitz spacetime in 21 (KLM)
Characterized by l , z . Reduces to AdS for z
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Isometry group
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Anisotropic holography
Key observation High-Low Temp.
duality (Holographic version)
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Key observation High-Low Temp. duality
(Holographic version)
Coordinate transformation
Both are diffeomorphic provided
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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
• 21 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.
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Solitons and the microscopic entropy of
asymptotically Lifshitz black holes
Euclidean action (Soliton)
Euclidean action (black hole)
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Euclidean action (black hole)
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Black hole entropy
Perfect matching provided
Field theory entropy
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An explicit example BHT Massive Gravity
E. A. Bergshoeff, O. Hohm, P. K. Townsend, PRL
2009
Lets focus on the special case
The theory admits Lifshitz spacetimes with
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An explicit example BHT Massive Gravity
Special case
Asymptotically Lifshitz black hole
E. Ayón-Beato, A. Garbarz, G. Giribet and M.
Hassaine, PRD 2009
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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 !

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Euclidean asymptotically Lifshitz black hole is
diffeomorphic to the gravitational soliton
Coordinate transformation
Followed by
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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)
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Regularized Euclidean action
Gravitational soliton
Finite action
Fixed mass
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Black hole (Can be obtained from the soliton
High Low Temp. duality)
Finite action
Black hole mass
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Black hole mass
Black hole entropy
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Black hole entropy (microcanonical ensemble)
Perfect matching with field theory entropy
(z 3) provided
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• Ending remarks Specific heat, phase
  transitions and an extension of cosmic
  censorship.

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Remarks

• 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

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Asymptotic growth of the number of states

• Canonical ensemble, 1st law

Reduces to Stefan-Boltzmann for z1