String-scale UV cut-off in the gravitational path integral removes string-coupling dependence from the BPS black hole index in extended supergravity, matching supersymmetry expectations for zero-Euler-number cases.
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Two extremal black holes with secondary scalar hair are identified in a two-U(1) Einstein-Maxwell-scalar theory using scalarization and entropy function methods, implying primary hair is difficult to obtain.
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Extended Supergravity Needs String Scale Cut-off
String-scale UV cut-off in the gravitational path integral removes string-coupling dependence from the BPS black hole index in extended supergravity, matching supersymmetry expectations for zero-Euler-number cases.