Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes

TL;DR

This paper explores field theories with anisotropic scaling in 1+1 dimensions, establishing a duality between low and high temperature regimes, and deriving a generalized Cardy formula for Lifshitz black holes without relying on asymptotic symmetries.

Contribution

It introduces a duality based on Lifshitz algebra isomorphism, providing a new way to compute black hole entropy in Lifshitz spacetimes without using asymptotic symmetries.

Findings

Derived a duality between low and high temperature regimes in Lifshitz theories.

Generalized Cardy formula for Lifshitz black holes incorporating dynamical exponent z.

Constructed a gravitational soliton solution matching black hole entropy for z=3.

Abstract

Field theories with anisotropic scaling in 1+1 dimensions are considered. It is shown that the isomorphism between Lifshitz algebras with dynamical exponents z and 1/z naturally leads to a duality between low and high temperature regimes. Assuming the existence of gap in the spectrum, this duality allows to obtain a precise formula for the asymptotic growth of the number of states with a fixed energy which depends on z and the energy of the ground state, and reduces to the Cardy formula for z=1. The holographic realization of the duality can be naturally inferred from the fact that Euclidean Lifshitz spaces in three dimensions with dynamical exponents and characteristic lengths given by z, l, and 1/z, l/z, respectively, are diffeomorphic. The semiclassical entropy of black holes with Lifshitz asymptotics can then be recovered from the generalization of Cardy formula, where the ground state corresponds to a soliton. An explicit example is provided by the existence of a purely gravitational soliton solution for BHT massive gravity, which precisely has the required energy that reproduces the entropy of the analytic asymptotically Lifshitz black hole with z=3. Remarkably, neither the asymptotic symmetries nor central charges were explicitly used in order to obtain these results.