Mixed Rademacher and BPS black holes

The exact degeneracies of quarter-BPS dyons in Type II string theory on K3 × T 2 are given by Fourier coefficients of the inverse of the Igusa cusp form. For a fixed magnetic charge invariant m, the generating function of these degeneracies naturally decomposes as a sum of two parts, which are supposed to account for single-centered black holes, and two-centered black hole bound states, respectively. The decomposition is such that each part is separately modular covariant but neither is holomorphic, calling for a physical interpretation of the non-holomorphy. We resolve this puzzle by computing the supersymmetric index of the quantum mechanics of two-centered half-BPS black-holes, which we model by geodesic motion on Taub-NUT space subject to a certain potential. We compute a suitable index using localization methods, and find that it includes both a temperature-independent contribution from BPS bound states, as well as a temperaturedependent contribution due to a spectral asymmetry in the continuum of scattering states. The continuum contribution agrees precisely with the non-holomorphic completion term required for the modularity of the generating function of two-centered black hole bound states.

Section: Introductionmentioning

confidence: 99%

Mock modularity from black hole scattering states

Murthy

Pioline

2018

“…[52,[60][61][62], but none of these studies employ the localization techniques used here. Indeed, even the untwisted quarter-BPS index is not completely understood from this perspective; see [57][58][59]63] for progress towards deriving and interpreting a Rademacher-type expansion for the quarter-BPS spectrum.…”

Section: Discussionmentioning

confidence: 99%

Exact half-BPS black hole entropies in CHL models from Rademacher series

Nally¹

2019

The microscopic spectrum of half-BPS excitations in toroidally compactified heterotic string theory has been computed exactly through the use of results from analytic number theory. Recently, similar quantities have been understood macroscopically by evaluating the gravitational path integral on the M-theory lift of the AdS 2 near-horizon geometry of the corresponding black hole. In this paper, we generalize these results to a subset of the CHL models, which include the standard compactification of IIA on K3 × T 2 as a special case. We begin by developing a Rademacher-like expansion for the Fourier coefficients of the partition functions for these theories, which are modular forms for congruence subgroups. We then interpret these results in a macroscopic setting by evaluating the path integral for the reduced-rank N = 4 supergravities described by these CFTs.

“…This indeed clarifies the modular nature of the degeneracies of black hole microstates, but we would like to do better and find an explicit formula for them, as in the N = 8 theory. Upon applying the circle method of Hardy-Ramanujan-Rademacher to the known modular completions of the mock Jacobi forms, one obtains an analytic formula for the black hole degeneracies [10] which is similar to, but more complicated than, the one in the N = 8 theory -there are some additional terms coming from the fact that one has mock Jacobi and not true Jacobi forms, but the bottom line is that for a given mock Jacobi form one has an infinite series of terms controlled purely by a finite number of integers. (The explicit formula is presented in (A.12).)…”

Section: Motivation and Contextmentioning

confidence: 99%

“…We review this in section 2. For the particular mock Jacobi forms ψ F m the formula was obtained in [10], which we recall in (A.12).…”

Section: The Main Formulamentioning

confidence: 99%

Dyonic black hole degeneracies in $$ \mathcal{N} $$ = 4 string theory from Dabholkar-Harvey degeneracies

Chowdhury

Kidambi

Murthy

et al. 2020

Self Cite

The degeneracies of single-centered dyonic $$ \frac{1}{4} $$ 1 4 -BPS black holes (BH) in Type II string theory on K3×T2 are known to be coefficients of certain mock Jacobi forms arising from the Igusa cusp form Φ10. In this paper we present an exact analytic formula for these BH degeneracies purely in terms of the degeneracies of the perturbative $$ \frac{1}{2} $$ 1 2 -BPS states of the theory. We use the fact that the degeneracies are completely controlled by the polar coefficients of the mock Jacobi forms, using the Hardy-Ramanujan-Rademacher circle method. Here we present a simple formula for these polar coefficients as a quadratic function of the $$ \frac{1}{2} $$ 1 2 -BPS degeneracies. We arrive at the formula by using the physical interpretation of polar coefficients as negative discriminant states, and then making use of previous results in the literature to track the decay of such states into pairs of $$ \frac{1}{2} $$ 1 2 -BPS states in the moduli space. Although there are an infinite number of such decays, we show that only a finite number of them contribute to the formula. The phenomenon of BH bound state metamorphosis (BSM) plays a crucial role in our analysis. We show that the dyonic BSM orbits with U-duality invariant ∆ < 0 are in exact correspondence with the solution sets of the Brahmagupta-Pell equation, which implies that they are isomorphic to the group of units in the order ℤ[$$ \sqrt{\left|\Delta \right|} $$ Δ ] in the real quadratic field ℚ($$ \sqrt{\left|\Delta \right|} $$ Δ ). We check our formula against the known numerical data arising from the Igusa cusp form, for the first 1650 polar coefficients, and find perfect agreement.