Supersymmetric near-horizon geometries in D = 6 supergravity: Lichnerowicz theorems, index theory and symmetry enhancement

Abstract

We analyse supersymmetric near-horizon geometries of extremal black holes in $N=(1,0)$, $D=6$ supergravity with one tensor multiplet and $U(1)$ $R$-symmetry gauging. Assuming smooth bosonic fields and a compact, connected, boundaryless spatial horizon section $\mathcal{S}$, we solve the Killing spinor equations (KSEs) along the lightcone directions and identify the independent horizon system satisfied by the spinors $\eta_\pm$ on $\mathcal{S}$. We then prove generalized Lichnerowicz-type theorems for both lightcone chiralities, showing that the zero modes of the relevant horizon Dirac operators are in one-to-one correspondence with Killing spinors on $\mathcal{S}$. As a consequence, the supersymmetry-counting formula $N = 2N_{-} + \mathrm{Index}(D_E)$ holds for the class of regular horizons under consideration, where $D_E$ is the horizon Dirac operator twisted by the bundle naturally associated to the gauge structure of the theory. The $D=6$ case is distinguished from the previously analysed $D=11$ and type-IIA horizons because $\mathcal{S}$ is a compact four-manifold and the theory is chiral, so the relevant index need not vanish. In the ungauged case this reduces to the ordinary chiral Dirac index on $\mathcal{S}$, while in the gauged case the index is that of the corresponding twisted operator. We also analyse the map $\eta_- \mapsto \Gamma_+\Theta_-\eta_-$. For non-trivial fluxes, the resulting spacetime $\mathfrak{sl}(2,\mathbb{R})$ symmetry is proved unconditionally in the ungauged theory. In the gauged theory the same conclusion follows provided one assumes $\mathrm{Ker}\,\Theta_- = \{0\}$. We state this assumption explicitly and do not claim a full gauged symmetry-enhancement theorem without it.