  The global U-dualities of extended supergravity have played a central role in
differentiating the distinct classes of extremal black hole solutions. When the
U-duality group satisfies certain algebraic conditions, as is the case for a
broad class of supergravities, the extremal black holes enjoy a further
symmetry known as Freudenthal duality (F-duality), which although distinct from
U-duality preserves the Bekenstein-Hawking entropy. Here it is shown that, by
adopting the doubled Lagrangian formalism, F-duality, defined on the doubled
field strengths, is not only a symmetry of the black hole solutions, but also
of the equations of motion themselves. A further role for F-duality is
introduced in the context of world-sheet actions. The Nambu-Goto world-sheet
action in any (t, s) signature spacetime can be written in terms of the F-dual.
The corresponding field equations and Bianchi identities are then related by
F-duality allowing for an F-dual formulation of Gaillard-Zumino duality on the
world-sheet. An equivalent polynomial "Polyakov- type" action is introduced
using the so-called black hole potential. Such a construction allows for
actions invariant under all groups of type E7, including E7 itself, although in
this case the stringy interpretation is less clear.
