Localization in Seven-Dimensional Supergravity

Seven-dimensional supergravity provides holographic access to strongly coupled six-dimensional superconformal theories through its highest-dimensional supersymmetric AdS vacua. Its unique spinor-bilinear geometry and massive higher-form dynamics make systematic analysis and solution-finding distinctively challenging. Relative equivariant localization can obtain supersymmetric observables in odd dimensions from fixed-point data on an auxiliary even-dimensional filling, without an explicit solution. We extend this approach to seven dimensions in maximal Cartan-gauged supergravity with all gauge fields active. We formulate the Euclidean action and Killing-spinor equations in terms of field strengths, and construct a bilinear system adapted to the change in spinor reality conditions from symplectic-Majorana to Majorana. On the non-null gauged active spinor branch, independent, oppositely R-charged spinors are paired by Majorana conjugation on the same complex background. One algebraic gauging correction and two algebraic bilinear projections suffice to extend the equal-charge sector to independent Cartan charges, and the underlying efficient reorganization also applies in the Lorentzian theory. Massive 3-form self-duality fixes the quartic anomaly class and determines the higher-form fixed-point datum from Abelian moment maps. Localization at the single fixed point of the filling recovers the general supersymmetric AdS$_7$ black-hole action, without extending the solution. We prove gauge-covariant Killing-spinor invariance under the combined supersymmetric Killing flow and R-symmetry action, deriving the linear chemical-potential constraint. Extremizing the constrained entropy functional and requiring real positive stationary entropy yields the nonlinear charge relation within the Euclidean analysis, equivalent to Lorentzian horizon extremality for the known spherical black-hole family.