The heterotic superpotential and the alpha' expansion

We study the role of the Hull connection in the heterotic superpotential for N=1 d=4 vacua. The integrand is defined in terms of H, which is in turn defined in terms of a gravitational Chern--Simons coupling involving H. We show this recursive structure is key to W reproducing some of the ten--dimensional supersymmetry conditions written down by Bergshoeff--de Roo through α'^3. With this evidence in hand we make some observations about extending some of the tensor equations of Hull and Strominger to all orders in α'.