Vacua of M theory and string theory

We argue that supersymmetric higher-dimension operators in the effective actions of M theory and type IIB string theory do not affect the maximally supersymmetric vacua: ${\mathrm{AdS}}_4\times S^7$ and ${\mathrm{AdS}}_7\times S^4$ in M theory and ${\mathrm{AdS}}_5\times S^5$ in type IIB string theory. All these vacua are described in superspace by a fixed point with all components of supertorsion and supercurvature being supercovariantly constant. This follows from 32 unbroken supersymmetries and allows us to prove that such vacua are exact.