Generalised Holonomy for Higher-Order Corrections to Supersymmetric Backgrounds in String and M-Theory

The notion of {\it generalised structure groups} and {\it generalised holonomy groups} has been introduced in supergravity, in order to discuss the spinor rotations generated by commutators of supercovariant derivatives when non-vanishing form fields are included, with their associated gamma-matrix structures that go beyond the usual \Gamma_{MN} of the Riemannian connection. In this paper we investigate the generalisations to the usual Riemannian structure and holonomy groups that result from the inclusion of higher-order string or M-theory corrections in the supercovariant derivative. Even in the absence of background form fields, these corrections introduce additional terms \Gamma_{M_1... M_6} in the supercovariant connection, and hence they lead to enlarged structure and holonomy groups. In some cases, the corrected equations of motion force form fields to become non-zero too, which can further enlarge the groups. Our investigation focuses on the generalised structure and holonomy groups in the transverse spaces K_n of (Minkowski) \times K_n backgrounds for n=6, 7, 8 and 10, and shows how the generalised holonomies allow the continued existence of supersymmetric backgrounds even though the usual Riemannian special holonomy is destroyed by the inclusion of the string or M-theory corrections.