Lifshitz scaling, microstate counting from number theory and black hole entropy
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Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation E ∼ kz 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.
KeywordsIntegrable Field Theories Space-Time Symmetries Classical Theories of Gravity Gauge-gravity correspondence
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- P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, NY, U.S.A. (1997) [INSPIRE].
- J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
- T.M. Apostol, Modular functions and Dirichlet series in number theory, Springer, New York, NY, U.S.A. (1990).Google Scholar
- G.H. Hardy and S. Ramanujan, Asymptotic formulæ in combinatory analysis, Proc. Lond. Math. Soc. s2-17 (1918) 75.Google Scholar
- E.M. Wright, Asymptotic partition formulae. III. Partitions into k-th powers, Acta Math. 63 (1934) 143.Google Scholar
- A. Gafni, Power partitions, J. Number Theor. 163 (2016) 19 [arXiv:1506.06124].
- Y.-L. Li and Y.-G. Chen, On the r-th root partition function, II, J. Number Theor. 188 (2018) 392.Google Scholar
- E. Lifshitz, On the theory of second-order phase transitions I, Zh. Eksp. Teor. Fiz. 11 (1941) 255.Google Scholar
- E. Lifshitz, On the theory of second-order phase transitions II, Zh. Eksp. Teor. Fiz. 11 (1941) 269.Google Scholar
- D. Melnikov, F. Novaes, A. Pérez and R. Troncoso, work in progress.Google Scholar
- Y. Matsuno, Bilinear transformation method, volume 174, Academic Press, New York, NY, U.S.A. (1984).Google Scholar
- A. Degasperis, D. Lebedev, M. Olshanetsky, S. Pakuliak, A. Perelomov and P. Santini, Nonlocal integrable partners to generalized MKdV and two-dimensional Toda lattice equations in the formalism of a dressing method with quantized spectral parameter, Commun. Math. Phys. 141 (1991) 133 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
- A. Das, Integrable models, World Sci. Lect. Notes Phys. 30 (1989) 1 [INSPIRE].
- P.J. Olver, Applications of Lie groups to differential equations, volume 107, Springer, U.S.A. (1986).Google Scholar
- A. Achucarro and P.K. Townsend, Extended supergravities in d = (2 + 1) as Chern-Simons theories, Phys. Lett. B 229 (1989) 383 [INSPIRE].
- E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].