Kerr effective black hole geometries in supergravity

We derive the explicit embedding of the effective Kerr spacetimes, which are pertinent to the vanishing of static Love numbers, soft hair descriptions of Kerr black holes, and low-frequency scalar-Kerr scattering amplitudes, as solutions within <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:math> supergravity. These spacetimes exhibit a hidden <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>S</c:mi><c:mi>L</c:mi><c:mo stretchy="false">(</c:mo><c:mn>2</c:mn><c:mo>,</c:mo><c:mi>R</c:mi><c:mo stretchy="false">)</c:mo><c:mo>×</c:mo><c:mi>U</c:mi><c:mo stretchy="false">(</c:mo><c:mn>1</c:mn><c:mo stretchy="false">)</c:mo></c:math> or <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>S</i:mi><i:mi>O</i:mi><i:mo stretchy="false">(</i:mo><i:mn>4</i:mn><i:mo>,</i:mo><i:mn>2</i:mn><i:mo stretchy="false">)</i:mo></i:math> symmetry resembling the so called subtracted geometries with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>S</m:mi><m:mi>L</m:mi><m:mo stretchy="false">(</m:mo><m:mn>2</m:mn><m:mo>,</m:mo><m:mi>R</m:mi><m:mo stretchy="false">)</m:mo><m:mo>×</m:mo><m:mi>S</m:mi><m:mi>L</m:mi><m:mo stretchy="false">(</m:mo><m:mn>2</m:mn><m:mo>,</m:mo><m:mi>R</m:mi><m:mo stretchy="false">)</m:mo></m:math> symmetry, which accurately represent the near-horizon geometry of Kerr black holes and, as we will argue most accurately represents the internal structure of the Kerr black hole. To quantify the differences among the effective Kerr spacetimes, we compare their physical quantities, internal structures, and geodesic equations. Although their thermodynamic properties, including entropy, match those of Kerr, our study uncovers significant differences in the interiors of these effective Kerr solutions. A careful examination of the internal structure of the spacetimes highlights the distinctions between various effective Kerr geometries and their quasinormal spectra.