Dynamical symmetry enhancement near N=2, D=4 gauged supergravity horizons

TL;DR

This paper proves that smooth horizons in 4D gauged N=2 supergravity with compact sections exhibit enhanced supersymmetry and SL(2,R) symmetry, with their geometry characterized by specific topological and differential equations.

Contribution

It establishes supersymmetry enhancement and symmetry properties of horizons, linking topological invariants to geometric and supersymmetric features in gauged supergravity.

Findings

Horizons preserve 2c_1(K)+4 supersymmetries.

Horizons with c_1(K)=0 have SL(2,R) symmetry and preserve 4 or 8 supersymmetries.

Spatial horizon sections are described by first order nonlinear ODEs.

Abstract

We show that all smooth Killing horizons with compact horizon sections of 4-dimensional gauged N=2 supergravity coupled to any number of vector multiplets preserve $2 c_1({\cal K})+4 \ell$ supersymmetries, where ${\cal K}$ is a pull-back of the Hodge bundle of the special K\"ahler manifold on the horizon spatial section. We also demonstrate that all such horizons with $c_1({\cal K})=0$ exhibit an SL(2,R) symmetry and preserve either 4 or 8 supersymmetries. If the orbits of the SL(2,R) 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 $c_1({\cal K})=0$ 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.