Single-centered black hole microstate degeneracies from instantons in supergravity

Abstract

We obtain holographic constraints on the microscopic degeneracies of black holes by computing the exact macroscopic quantum entropy using localization, including the effects of string worldsheet instantons in the supergravity effective action. For 1 4 -BPS black holes in type II string theory on K3 × T 2 , the constraints can be explicitly checked against expressions for the microscopic BPS counting functions that are known in terms of certain mock modular forms. We find that the effect of including the infinite sum over instantons in the holomorphic prepotential of the supergravity leads to a sum over Bessel functions with successively sub-leading arguments as in the Rademacher expansion of Jacobi forms -but begins to disagree with such a structure near an order where the mock modular nature becomes relevant. This leads to a systematic method to recover the polar terms of the microscopic degeneracies from the degeneracy of instantons (the Gromov-Witten invariants). We check explicitly that our formula agrees with the known microscopic answer for the first seven values of the magnetic charge invariant.

Siegel modular forms and black hole entropy

“…It is interesting to note that for χ 10 one can take into account the exclusion principle in [47] by studying instead (4.11). In [48,49] the result for the microscopic degeneracy is shown to be a finite sum of Bessel functions, which is suggestive of a Rademacher expansion. As a matter of fact, the factors that multiply the Bessel functions can be identified with the polar coefficients [49] in a mock-Jacobi expansion [27].…”

“…In [48,49] the result for the microscopic degeneracy is shown to be a finite sum of Bessel functions, which is suggestive of a Rademacher expansion. As a matter of fact, the factors that multiply the Bessel functions can be identified with the polar coefficients [49] in a mock-Jacobi expansion [27]. Since under spectral flow from R to NS sector, the polar state degeneracy is mapped to the counting of chiral primaries, one can use the residue formula to make interesting predictions about the spectrum of KK fields of the dual bulk theory.…”

Mixed Rademacher and BPS black holes

“…It was shown in [27] that localizing the quantum entropy function with the prepotential (2.13) yields a sum of I-Bessel functions of weight 23/2, along with additional contributions that were dubbed "edge-effects". A comparison in the I-Bessel 23/2 sector was conducted for the cases m = 1 .…”

“…In addition, the measure of the localizing manifold [dφ I ] used in [27] was inspired by a saddle-point approximation of the Igusa cusp form derived in [12]. Using such a set-up, the localized quantum entropy function takes the form [27],…”

“…20 The precise contribution is recalled in appendix C, where some errors in [27] have been corrected.…”