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.

arXiv comment29 pages. v3: title updated, abstract and introduction rewritten, index of horizon Dirac operator evaluated explicitly via Atiyah-Singer theorem, results restructured with formal theorems/propositions/corollaries, gauged and ungauged cases carefully separated throughout

References in corpus (30)

• Large N Field Theories, String Theory and Gravity
• All supersymmetric solutions of minimal supergravity in five dimensions
• Supersymmetry and Attractors
• D--branes and Spinning Black Holes
• Vacuum interpolation in supergravity via super p-branes
• Near-horizon symmetries of extremal black holes
• A supersymmetric black ring
• Superrotation Charge and Supertranslation Hair on Black Holes
• Towards a Naturally Small Cosmological Constant from Branes in 6D Supergravity
• A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
• Supertranslations and Superrotations at the Black Hole Horizon
• Quantum Entropy Function from AdS(2)/CFT(1) Correspondence
• Rotating Attractors
• Higher dimensional black holes and supersymmetry
• Supersymmetric multi-charge AdS_5 black holes
• On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon
• Codimension Two Compactifications and the Cosmological Constant Problem
• Sidestepping the Cosmological Constant with Football-Shaped Extra Dimensions
• New Couplings of Six-Dimensional Supergravity
• Enhancement of Supersymmetry Near 5d Black Hole Horizon
• Tensor and Vector Multiplets in Six-Dimensional Supergravity
• All Couplings of Minimal Six-dimensional Supergravity
• On rigidity of analytic black holes
• Supergravity vacua and lorentzian Lie groups
• Symmetries at the black hole horizon
• Index theory and dynamical symmetry enhancement near IIB horizons
• Dynamical symmetry enhancement near IIA horizons
• Dynamical symmetry enhancement near massive IIA horizons
• Dynamical symmetry enhancement near N=2, D=4 gauged supergravity horizons
• Symmetry enhancement of extremal horizons in D=5 supergravity