From BPS to Non-BPS Black Holes Canonically

TL;DR

The paper shows that double-extremal black holes form an integrable system in action-angle variables, enabling a canonical transformation to relate BPS and non-BPS attractors. It develops a Kamiltonian framework for the reduced one-dimensional dynamics, connects it to Legendre/Hesse duality, and demonstrates explicit STU and $E_{7(7)}$ examples where BPS solutions map to non-BPS solutions without changing the black hole entropy. The work clarifies how spontaneous supersymmetry breaking is encoded in attractor equations and invariants, and it provides a unified, canonical perspective on extremal black holes across different supergravity theories. This canonical map offers a structured route to generate non-BPS solutions from BPS seeds and highlights the role of symplectic invariants in preserving entropy across the transformation.

Abstract

We describe the ``action-angle'' integrable system underlying the structure of double-extremal black holes. This implies the existence of a canonical transformation from BPS to non-BPS black holes. We give examples of such canonical transformation for STU and for E(7(7))-invariant black holes