Abstract
We consider four-dimensional dyonic single-center BPS black holes in the N = 2 STU model of Sen and Vafa. By working in a region of moduli space where the real part of two of the three complex scalars S, T , U are taken to be large, we evaluate the quantum entropy function for these BPS black holes. In this regime, the subleading corrections point to a microstate counting formula partly based on a Siegel modular form of weight two. This is supplemented by another modular object that takes into account the dependence on Y0, a complex scalar field belonging to one of the four off-shell vector multiplets of the underlying supergravity theory. We also observe interesting connections to the rational Calogero model and to formal deformation of a Poisson algebra, and suggest a string web picture of our counting proposal.
Highlights
A highly active area of focused research in string theory aims at a formulation of the microscopic degeneracy of states of black holes to arrive at a statistical description of black hole thermodynamics
We address the problem of black hole microstate counting for BPS asymptotically flat black hole backgrounds
As we review in appendix C, the In, with n ≥ 3, can be expressed in terms of 1st Rankin-Cohen brackets for modular forms, while I2 can be expressed in terms of the quasi-modular form I1 by making use of 1st Rankin-Cohen brackets for quasi-modular forms of depth
Summary
A highly active area of focused research in string theory aims at a formulation of the microscopic degeneracy of states of black holes to arrive at a statistical description of black hole thermodynamics. We find that ω(S), which is proportional to ln θ82(S), together with subleading corrections in the quantum entropy function, point to a dependence of the microstate counting formula on a Siegel modular form Φ2 of weight 2 This Siegel modular form can be constructed by applying a Hecke lift [5] to a specific Jacobi form constructed from the seed θ82, and it differs from the Siegel modular form proposed in [22], which did not take into account the subleading corrections just mentioned. It is based on the Siegel modular form Φ2 as well as on another modular object that captures the dependence on the complex scalar Y 0. In appendices A–G, we collect results about modular forms for SL(2, Z) and Γ0(2), Jacobi forms, Siegel modular forms, Rankin-Cohen brackets and Hecke lifts
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