Nonlinear symmetries of black hole entropy in gauged supergravity

Freudenthal duality in N=2, D=4 ungauged supergravity is generated by an anti-involutive operator that acts on the electromagnetic fluxes, and results to be a symmetry of the Bekenstein-Hawking entropy. We show that, with a suitable extension, this duality can be generalized to the abelian gauged case as well, even in presence of hypermultiplets. By defining Freudenthal duality along the scalar flow, one can prove that two configurations of charges and gaugings linked by the Freudenthal operator share the same set of values of the scalar fields at the black hole horizon. Consequently, Freudenthal duality is promoted to a nonlinear symmetry of the black hole entropy. We explicitly show this invariance for the model with prepotential $F=-i X^0 X^1$ and Fayet-Iliopoulos gauging.