Abstract
We propose a geometric method to study the residual symmetries in $N=2$, $d=4$ $\text{U}(1)$ Fayet-Iliopoulos (FI) gauged supergravity. It essentially involves the stabilization of the symplectic vector of gauge couplings (FI parameters) under the action of the U-duality symmetry of the ungauged theory. In particular we are interested in those transformations that act non-trivially on the solutions and produce scalar hair and dyonic black holes from a given seed. We illustrate the procedure for finding this group in general and then show how it works in some specific models. For the prepotential $F=-iX^0X^1$, we use our method to add one more parameter to the rotating Chow-Comp\`ere solution, representing scalar hair.
Highlights
Duality transformations have been instrumental in the construction of black hole solutions in string theory
When global symmetries of some given supergravity theory are gauged, as it typically happens in AdS supergravity, the sigma model target space isometries are generically broken by the presence of a scalar potential, so that the powerful solutiongenerating techniques described above seem to break down
In this paper we presented a geometric method to determine the residual symmetries in N = 2, d = 4 U(1) Fayet-Iliopoulos gauged supergravity
Summary
The scalar potential generically spoils this invariance, but, as is clear from (2.11), for dyonic gauging one recovers the whole U-duality invariance, at the price of changing the vector of gauge couplings and so the physical theory. We will call this group Ufi, that stands for fake internal symmetry group.. There are some cases in which Ui strictly contains SG, and this depends on some particular symmetric structures of the model under consideration This happens because the symmetry of the model allows to act with some symplectic matrices in a more general way than (2.13), leaving the theory invariant
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