A second rotational Killing field on gauged $D=5$ vector-multiplet horizons, and a no-go for varying-moduli black rings

Abstract

We study supersymmetric near-horizon geometries of gauged $D=5$ supergravity coupled to vector multiplets, on the branch where the canonical rotational Killing vector $\tilde{V}$ of the cross-section $\mathcal{S}$ is non-vanishing. No rotational symmetry is assumed, and nothing about the set where the frame built from the Killing spinors degenerates. On a compact connected $\mathcal{S}$ without boundary a second rotational Killing field, independent of $\tilde{V}$, always exists and is an isometry of all of $\mathcal{S}$. Where the moduli vary it is $$U_i=\parallel {\eta }_{-}{\parallel }^2(\alpha Z_i-{\epsilon }_{ijk}{Z}^j{u}^k),u_i=\Phi P_i-h_i,$$ a polynomial in the horizon data, hence smooth everywhere; where the moduli are constant the horizon is locally homogeneous. The only further hypothesis for these results is that the superpotential $\Phi =\chi V_I{X}^I$ is nowhere zero --- weaker than the non-negativity of the scalar potential assumed in the earlier literature. The two sub-branches are separated by $K=Q_{IJ}{C}^I{C}^J$, which vanishes exactly in the minimal theory: $K\equiv 0$ recovers the result of Grover, Gutowski, Papadopoulos and Sabra, while elsewhere $K>0$ and $\alpha$ is either identically zero or nowhere zero. Each of $K\equiv 0$, $P\equiv 0$ and $P\not\equiv 0$ occurs on compact $\mathcal{S}$. Constant moduli return the local geometries of Kunduri and Lucietti as a conclusion, not an ansatz. Varying moduli with $\alpha \not\equiv 0$ give a cohomogeneity-one $T^2$ action whose orbit space is a closed interval, so $\mathcal{S}$ is $S^3$, a lens space or $S^1\times S^2$; the last is excluded by two global first integrals, a new one, $\alpha \parallel {\eta }_{-}{\parallel }^4$, and the constant spinor norm already known. A varying-moduli supersymmetric $Ad{S}_5$ black ring therefore cannot exist with $\alpha \not\equiv 0$; the $S^1\times S^2$ window survives only at constant moduli or at $\alpha \equiv 0$.

91 pages, no figures. v2: the second Killing field extends to an unconditional global isometry via an explicit frame-free closed form (new Sec. 7.2). A cohomogeneity-one toric reduction of the varying-moduli branch (new Sec. 7.7) excludes the S^1 x S^2 (black-ring) topology when alpha is not identically zero. Charge and hidden-symmetry results and a data availability statement are also added