Generalized holonomy of supergravities with 8 real supercharges*1

Abstract

We show that the generalized holonomy groups of ungauged supergravity theories with 8 real supercharges must be contained in SL(2−ν,H)⋉νH2−ν⊆SL(2,H), where SL(2,H) is the generalized structure group. Here n=4ν is the number of preserved supersymmetries, so the allowed values are limited to n=0,4,8. In particular, solutions of ungauged supergravities in four, five and six dimensions are examined and found to explicitly follow this pattern. We also argue that the G-structure has to be a subgroup of this generalized holonomy group, which may provide a possible classification for supergravity vacua with respect to the number of supercharges.