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Field theories with anisotropic scaling in 1+1 dimensions are considered. Itis shown that the isomorphism between Lifshitz algebras with dynamicalexponents z and 1/z naturally leads to a duality between low and hightemperature regimes. Assuming the existence of gap in the spectrum, thisduality allows to obtain a precise formula for the asymptotic growth of thenumber of states with a fixed energy which depends on z and the energy of theground state, and reduces to the Cardy formula for z=1. The holographicrealization of the duality can be naturally inferred from the fact thatEuclidean Lifshitz spaces in three dimensions with dynamical exponents andcharacteristic lengths given by z, l, and 1/z, l/z, respectively, arediffeomorphic. The semiclassical entropy of black holes with Lifshitzasymptotics can then be recovered from the generalization of Cardy formula,where the ground state corresponds to a soliton. An explicit example isprovided by the existence of a purely gravitational soliton solution for BHTmassive gravity, which precisely has the required energy that reproduces theentropy of the analytic asymptotically Lifshitz black hole with z=3.Remarkably, neither the asymptotic symmetries nor central charges wereexplicitly used in order to obtain these results.

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