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

TL;DR

This paper investigates supersymmetric near-horizon geometries in six-dimensional supergravity, establishing new theorems relating Killing spinors to Dirac operator zero modes, and analyzing symmetry enhancements with or without gauge fluxes.

Contribution

It proves generalized Lichnerowicz theorems for D=6 supergravity horizons, relates supersymmetry to Dirac operator indices, and discusses conditions for symmetry enhancement.

Findings

Zero modes of horizon Dirac operators correspond to Killing spinors.

Supersymmetry count follows from index theory of twisted Dirac operators.

Unconditional $ ext{sl}(2, ext{R})$ symmetry in ungauged case; conditional in gauged case.

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.