Choices of spinor inner products on M-theory backgrounds

Abstract

M-theory backgrounds in the form of unwarped compactifications with or without fluxes are considered. We construct the bilinear forms of supergravity Killing spinors for different choices of spinor inner products on these backgrounds. The equations satisfied by the bilinear forms and their decompositions into product manifolds are obtained for different inner product choices in the special case for which the spinors factorize. It is found that the $AdS$ solutions can only appear for some special choices of spinor inner products on product manifolds. The reduction of bilinears of supergravity Killing spinors into the hidden symmetries of product manifolds which are Killing-Yano and closed conformal Killing-Yano forms for $AdS$ solutions is shown. These hidden symmetries are lifted to eleven-dimensional backgrounds to find the hidden symmetires on them. The relation between the choices of spinor inner products, $AdS$ solutions and hidden symmetries on M-theory backgrounds are investigated.