Abstract
Higher spin gravity in three dimensions has explicit black holes solutions, carrying higher spin charge. We compute the free energy of a charged black hole from the holographic dual, a 2d CFT with extended conformal symmetry, and find exact agreement with the bulk thermodynamics. In the CFT, higher spin corrections to the free energy can be calculated at high temperature from correlation functions of W-algebra currents.
Highlights
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These black holes encode the thermodynamics of the CFT at high temperature, which in two dimensions is determined by the chiral algebra
As we have shown in this paper, we can reproduce the higher spin corrections to the black hole entropy from integrals of correlation functions of W-currents on the torus
Summary
Let us begin by explaining the aim of our calculation in more detail. The entropy calculation of [26] amounts, from the dual CFT point of view, to the evaluation of the partition function. The bulk partition function (1.1) was obtained in [26] from the thermodynamics of the black hole solution in AdS3 higher spin gravity It is an expansion in powers of the chemical potential α, so it should be compared to the CFT expansion (2πiα). As is familiar from entropy calculations [28], the standard method to obtain the partition function from a dual conformal field theory point of view is to do the S-modular transformation. Where the sum runs over all primaries labelled by r, s (with r = 0 the vacuum representation), Ssr is the modular S-matrix (not to be confused with the black hole entropy), and the dots indicate terms exponentially suppressed at high temperature. There have been some ideas that the relevant transformation property will be of the same structure as that described in [30], see [31], but no general formula is known. As we shall explain in the following, we can derive the transformation properties of these traces from first principles
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