Black holes and general Freudenthal transformations

Abstract

Abstract We study General Freudenthal Transformations (GFT) on black hole solutions in Einstein-Maxwell-Scalar (super)gravity theories with global symmetry of type E 7. GFT can be considered as a 2-parameter, a, b ∈ ℝ, generalisation of Freudenthal duality: $$ x\to {x}_F= ax+b\tilde{x} $$ x → x F = ax + b x ˜ , where x is the vector of the electromagnetic charges, an element of a Freudenthal triple system (FTS), carried by a large black hole and $$ \tilde{x} $$ x ˜ is its Freudenthal dual. These transformations leave the Bekenstein-Hawking entropy invariant up to a scalar factor given by a 2 ± b 2. For any x there exists a one parameter subset of GFT that leave the entropy invariant, a 2 ± b 2 = 1, defining the subgroup of Freudenthal rotations. The Freudenthal plane defined by spanℝ{x, $$ \tilde{x} $$ x ˜ } is closed under GFT and is foliated by the orbits of the Freudenthal rotations. Having introduced the basic definitions and presented their properties in detail, we consider the relation of GFT to the global symmetries or U-dualities in the context of supergravity. We consider explicit examples in pure supergravity, axion-dilaton theories and N = 2, D = 4 supergravities obtained from D = 5 by dimensional reductions associated to (non-degenerate) reduced FTS’s descending from cubic Jordan Algebras.