Scaling black holes and modularity

A bstract Scaling black holes are solutions of supergravity with multiple hole singularities, which can be adiabatically connected to a single center solution. We develop techniques determine partition functions for such scaling holes, if each constituent carries non-vanishing magnetic charge corresponding D4-brane in string theory, or equivalently M5-brane M-theory. For three constituents, we demonstrate that the function is mock modular form depth two, and appropriate non-holomorphic completion using generalized error functions. From four-dimensional perspective, parameter axion-dilaton, our results show S -duality leaves this subset spectrum invariant. five-dimensional complex structure torus T 2 , dual states dimensional reduction worldvolume theory . As case study, specialize compactification manifold K3 fibration, explicitly evaluate holomorphic parts