Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy

Abstract

Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation $E\sim k^z$ 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 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 $\zeta$-function. The quantum Benjamin-Ono${}_2$ (BO${}_2$) 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 AdS${}_3$ is described by the BO${}_2$ equations. This suggests that the phase space can be quantized in terms of quantum BO${}_2$ states. Indeed, in the semiclassical limit, the ground state energy of BO${}_2$ coincides with the energy of global AdS${}_3$, and the Bekenstein-Hawking entropy for BTZ black holes is recovered from the anisotropic extension of Cardy formula.

21 pages. v2: minor corrections + extended discussion on the generalized modular invariance in the intro; matches published version

References in corpus (10)

• On space of integrable quantum field theories
• $T\bar{T}$-deformed 2D Quantum Field Theories
• Cutoff AdS${}_3$ versus the $T\bar{T}$ deformation
• Modular invariance and uniqueness of $T\bar{T}$ deformed CFT
• $T\bar{T}$ deformed partition functions
• Generalized Black Holes in Three-dimensional Spacetime
• Quantum Shock Waves - the case for non-linear effects in dynamics of electronic liquids
• Integrable hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation
• Integrable systems with BMS${}_3$ Poisson structure and the dynamics of locally flat spacetimes
• Log corrections to entropy of three dimensional black holes with soft hair

Cited by in corpus (15)

• Symmetry-Resolved Entanglement in AdS${}_3$/CFT${}_2$ coupled to $U(1)$ Chern-Simons Theory
• Charged Moments in $W_3$ Higher Spin Holography
• A slow review of the AGT correspondence
• Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies
• Gravitational dual of averaged free CFT's over the Narain lattice
• Integrable systems and the boundary dynamics of higher spin gravity on AdS${}_3$
• Scalar fields and Lifshitz black holes from Derrick's theorem evasion
• Gravity coupled to a scalar field from a Chern-Simons action: describing rotating hairy black holes and solitons with gauge fields
• BTZ black hole with KdV-type boundary conditions: Thermodynamics revisited
• The anisotropic chiral boson
• Charged Lifshitz black holes from general covariance breaking
• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• Universal sectors of two-dimensional Carrollian CFTs
• 1/c deformations of AdS${}_3$ boundary conditions and the Dym hierarchy
• Modular Properties of Generalised Gibbs Ensembles