Localization in Seven-Dimensional Supergravity
Abstract
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 AdS7 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.
Related
- Timelike Supersymmetric Geometries in Seven-Dimensional Supergravity
- All-Disorder Localization for the One-Dimensional Anderson Model with Power-Law Hopping
- Localization Lengths for Two-Dimensional Random Band Matrices: Stretched-Exponential Lower Bounds
- Componentwise and Measure-Sensitive Bounds for One-Dimensional Time-Frequency Localization
- Scale-free Hankel factorization and explicit one-dimensional plunge bounds for time-frequency localization operators
BibTeX
@misc{gaar2026localization,
title = {{Localization in Seven-Dimensional Supergravity}},
author = {Florian Gaar and Michèle Levi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40514},
eprint = {2610.40514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40514},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Localization in Seven-Dimensional Supergravity
AU - Florian Gaar
AU - Michèle Levi
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40514
UR - https://arxiv.org/abs/2610.40514
AB - 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.
ER -