Scaling Black Holes and Modularity
arXiv:2110.05504 · doi:10.1007/JHEP03(2022)001
Abstract
Scaling black holes are solutions of supergravity with multiple black hole singularities, which can be adiabatically connected to a single center black hole solution. We develop techniques to determine partition functions for such scaling black holes, if each constituent carries a non-vanishing magnetic charge corresponding to a D4-brane in string theory, or equivalently M5-brane in M-theory. For three constituents, we demonstrate that the partition function is a mock modular form of depth two, and we determine the appropriate non-holomorphic completion using generalized error functions. From the four-dimensional perspective, the modular parameter is the axion-dilaton, and our results show that -duality leaves this subset of the spectrum invariant. From the five-dimensional perspective, the modular parameter is the complex structure of a torus , and the scaling black holes are dual to states in the dimensional reduction of the M5-brane worldvolume theory to . As a case study, we specialize the compactification manifold to a K3 fibration, and explicitly evaluate holomorphic parts of partition functions.
arXiv comment52 pages
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