Connections, Field Redefinitions and Heterotic Supergravity

We study heterotic supergravity at $\mathcal{O}(\alpha')$, first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss an ambiguity of a connection choice on the tangent bundle. It is well known that at $\mathcal{O}(\alpha')$ the Hull connection gives a consistent supergravity theory with supersymmetry transformations given in the usual way. We consider deformations of this connection corresponding to field redefinitions, and the necessary corrections to the supersymmetry transformations. 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 $M_4\times X$, where $M_4$ is Minkowski and $X$ is compact, the moduli space of infinitesimal field redefinitions is given by $H^{(0,1)}(X,\textrm{End}(TX))$. 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 argue that this symmetry should be kept to higher orders in $\alpha'$, and propose a natural choice for the corresponding tangent bundle connection used in curvature computations. In particular, the Hull connection should be corrected at second and higher orders in $\alpha'$ from this point of view.