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Lifshitz scaling, microstate counting from number theory and black hole entropy

  • Dmitry Melnikov
  • Fábio NovaesEmail author
  • Alfredo Pérez
  • Ricardo Troncoso
Open Access
Regular Article - Theoretical Physics
  • 13 Downloads

Abstract

Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation Ekz and dynamical exponent z > 1. The asymptotic growth of the number of states of these theories can be described by an extension of Cardy formula that depends on z. We show that this result can be recovered by counting the partitions of an integer into z-th powers, as proposed by Hardy and Ramanujan a century ago. This gives a novel duality relationship between the characteristic energy of the dispersion relation with the cylinder radius and the ground state energy. For free bosons with Lifshitz scaling, this relationship is shown to be identically fulfilled by virtue of the reflection property of the Riemann ζ-function. The quantum Benjamin-Ono2 (BO2) integrable system, relevant in the AGT correspondence, is also analyzed. As a holographic realization, we provide a special set of boundary conditions for which the reduced phase space of Einstein gravity with a couple of U (1) fields on AdS3 is described by the BO2 equations. This suggests that the phase space can be quantized in terms of quantum BO2 states. Indeed, in the semiclassical limit, the ground state energy of BO2 coincides with the energy of global AdS3, and the Bekenstein-Hawking entropy for BTZ black holes is recovered from the anisotropic extension of Cardy formula.

Keywords

Integrable Field Theories Space-Time Symmetries Classical Theories of Gravity Gauge-gravity correspondence 

Notes

Open Access

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Dmitry Melnikov
    • 1
    • 2
  • Fábio Novaes
    • 1
    Email author
  • Alfredo Pérez
    • 3
  • Ricardo Troncoso
    • 3
  1. 1.International Institute of PhysicsFederal University of Rio Grande do NorteNatalBrazil
  2. 2.Institute for Theoretical and Experimental PhysicsMoscowRussia
  3. 3.Centro de Estudios Científicos (CECs)ValdiviaChile

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