New Near Extremal Black Holes and Love Symmetry

Rotating and charged black holes are known to exhibit remarkable properties close to extremality, including emergent hidden symmetries and holographic duality to 2D theories. In this note, we introduce a new class of near-extremal black holes living in $(2,2)$ signature, strongly resembling the Lorentzian ones but with an exact integrable structure $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$. The exterior of the black hole is a self-dual solution with a photon ring. It develops an infinite near-horizon throat described by the Eguchi-Hanson instanton. An exact version of the so-called Love symmetry controls perturbations on the throat. Furthermore, the normalizable spectrum of this black hole provides quasinormal modes in Lorentzian signature.