Partition function of non-supersymmetric black holes in the supergravity limit
In this paper, we propose a free energy for general non-supersymmetric black hole attractors (with vanishing D0-brane charge) arising in type IIA(B) superstrings on 3-fold Calabi–Yau, in the supergravity limit. This, by definition, differs from its counterpart BPS free energy by a factor of 4. Correspondingly, a mixed ensemble for these black holes is proposed.
The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational cost. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures: in particular we investigate the mutual benefit of combining in various ways suitable preconditioners with V-cycle algorithms. Numerical experiments in one and two spatial dimensions for the validation of our multi-facet analysis complement this contribution.