Exact BPS black hole microstate counting from holographic conformal quantum mechanics

In this note, we develop a prescription for describing BPS black hole microstates in terms of a holographic conformal quantum mechanics (CQM) model dual to the near-horizon $AdS_2$ geometry of the black hole. We use 1/2 BPS small black holes in a 4D ${\cal N}=4$ toroidal heterotic compactification as well as 1/8 BPS large black holes in a 4D ${\cal N}=8$ Type II toroidal compactification as test cases for our approach. In each case, the $SL(2,\mathbb{Z})$ modular symmetries of the known generating function of the exact microstate degeneracies enables the latter to be expressed as a Rademacher series expansion, with each summand consisting of phases and a modified Bessel function of the first kind. We make a motivated ansatz that the de Alfaro-Fubini-Furlan model (DFF) coupled to a simple harmonic oscillator is a universal sector of the holographic CQM dual to the BPS black hole's near-horizon $AdS_2$ geometry, and demonstrate how in both cases, the two parameters of this putative CQM, the DFF coupling as well as the oscillator frequency, exactly encode both the index and the argument of the Bessel function. Consequently, we extract the leading, logarithmic and all sub-leading power law black hole entropy contributions from calculations in the CQM. In the ${\cal N}=4$ case, the DFF ansatz is sufficient to successfully reproduce the exact microscopic generating function from the CQM.