Connections, field redefinitions and heterotic supergravity

We study heterotic supergravity at O(α^) , first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss the ambiguity of a connection choice on the tangent bundle. It is well known that in order to have a consistent supergravity with supersymmetry transformations given in the usual way, this connection must be the Hull connection at O(α^) . We consider deformations of this connection corresponding to field redefinitions, and the necessary corrections to the supersymmetry transformations. We also discuss possible extensions of this theory to higher orders in α′. We are interested in the moduli space of such field redefinitions which allow for supersymmetric solutions to the equations of motion. We show that for solutions on M 4 × X, where M 4 is Minkowski and X is compact, this is given by H (0,1)(X, End(TX)). This space corresponds to infinitesimally close connections for which the equations of motion are satisfied. The setup suggests a symmetry between the gauge connection and the tangent bundle connection, as also employed by Bergshoeff and de Roo. We propose that this symmetry should be kept to higher orders in α′, and propose a natural choice for the corresponding tangent bundle connection used in curvature computations.