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Anisotropic holography and the microscopic entropy

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 arXiv:1107.3647 [ hep-th ].

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Anisotropic holography and the microscopic entropy

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  1. Anisotropic holography and the microscopic entropy of Lifshitzblackholes in 3D Ricardo Troncoso In collaborationwith 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 D : P : H : Two-dimensional Lifshitz algebra with dynamical exponent z :

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

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

  5. On a cylinder : Finitetemperature (torus) :

  6. 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 !

  7. 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 :

  8. 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 :

  9. Asymptotic growth of the number of states Note that for z=1 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 !

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

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

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

  14. Anisotropicholography: Solitons and themicroscopicentropy of asymptoticallyLifshitzblackholes • 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 themicroscopicentropy of asymptoticallyLifshitzblackholes Euclidean action (Soliton) : Euclidean action (black hole) :

  16. Euclidean action (black hole) :

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

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

  19. Anexplicitexample : BHT MassiveGravity Special case : Asymptotically Lifshitz black hole : E. Ayón-Beato, A. Garbarz, G. Giribet and M. Hassaine, PRD 2009

  20. Anexplicitexample : BHT MassiveGravity 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 !

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

  22. RegularizedEuclideanaction 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)

  23. RegularizedEuclideanaction 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 mass : Black hole entropy :

  26. Black holeentropy (microcanonicalensemble) Perfect matching with field theory entropy (z = 3) provided

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

  28. 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

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

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