1. Anisotropic holography and the microscopic entropy of Lifshitz black holes in 3D of Lifshitz black holes in 3D Ricardo Troncoso Ricardo Troncoso In collaboration with In collaboration with Hernán González and David Tempo Hernán González and David Tempo Centro de Estudios Científicos (CECS) Valdivia, Chile arXiv:1107.3647 [hep-th]

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

3. Isomorphism : This isomorphism induces the equivalence of Z between low and high T Key observation

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

5. Finite temperature (torus) : On a cylinder :

6. Field theories with anisotropic scaling in 2d (finite 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 !

7. 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 states

8. 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

9. * 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 ! Asymptotic growth of the number of states Note that for z=1 reduces to Cardy formula *

10. 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

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

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

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

14. 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.

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

16. Euclidean action (black hole) :

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

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

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

20. Special case : An explicit example : BHT Massive Gravity 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 !

21. Euclidean asymptotically Lifshitz black hole is diffeomorphic to the gravitational soliton : Coordinate transformation : Followed by :

22. Regularized Euclidean action Regularization intended for the black hole with z = 3, l It must necessarily work for the soliton ! (z = 1/3, l/3) O. Hohm and E. Tonii, JHEP 2010

23. Regularized Euclidean action Gravitational soliton : Finite action : Fixed mass :

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

25. Black hole entropy : Black hole mass :

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

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

29. 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 :

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