Dynamical symmetry enhancement near $$ \mathcal{} $$ = 2, D = 4 gauged supergravity horizons

We show that all smooth Killing horizons with compact horizon sections of 4-dimensional gauged \( \mathcal{N} \) = 2 supergravity coupled to any number of vector multiplets preserve \( 2{c}_1\left(\mathcal{K}\right)+4\ell \) supersymmetries, where \( \mathcal{K} \) is a pull-back of the Hodge bundle of the special Kähler manifold on the horizon spatial section. We also demonstrate that all such horizons with \( {c}_1\left(\mathcal{K}\right)=0 \) exhibit an \( \mathfrak{s}\mathfrak{l}\left(2,\mathbb{R}\right) \) symmetry and preserve either 4 or 8 supersymmetries. If the orbits of the \( \mathfrak{s}\mathfrak{l}\left(2,\mathbb{R}\right) \) symmetry are 2-dimensional, the horizons are warped products of AdS2 with the horizon spatial section. Otherwise, the horizon section admits an isometry which preserves all the fields. The proof of these results is centered on the use of index theorem in conjunction with an appropriate generalization of the Lichnerowicz theorem for horizons that preserve at least one supersymmetry. In all \( {c}_1\left(\mathcal{K}\right)=0 \) cases, we specify the local geometry of spatial horizon sections and demonstrate that the solutions are determined by first order non-linear ordinary differential equations on some of the fields.