On the supersymmetries of anti de Sitter vacua

TL;DR

The paper develops a geometric method to assign a Lie superalgebra to AdS_p+2 × X bosonic vacua by combining Killing vectors, Killing spinors, the spinorial Lie derivative, and spinor bilinears, together with the cone correspondence to reduce spinor data to group-theoretic data. It shows how Nahm’s classification yields unique symmetry superalgebras for AdS_4 × X_7, AdS_5 × X_5, and AdS_7 × X_4 in cases where X are Sasaki–Einstein or 3-Sasaki, clarifying when explicit spinors are not needed. The approach unifies and extends prior work by enabling group-theoretical determination of the symmetry content using holonomy data of the cone. This provides a robust geometric foundation for understanding AdS/CFT symmetries in a broad class of M-theory and string-theory vacua. The results offer precise symmetry algebras consistent with existing classifications and illuminate the role of Sasaki–Einstein and 3-Sasaki geometries in AdS supersymmetry.

Abstract

We present details of a geometric method to associate a Lie superalgebra with a large class of bosonic supergravity vacua of the type AdS x X, corresponding to elementary branes in M-theory and type II string theory.