  We study the four-dimensional effective theory arising from ten-dimensional
heterotic supergravity compactified on manifolds with torsion. In particular,
given the heterotic superpotential appropriately corrected at
$\mathcal{O}(\alpha')$ to account for the Green-Schwarz anomaly cancellation
mechanism, we investigate properties of four-dimensional Minkowski vacua of
this theory. Considering the restrictions arising from F-terms and D-terms we
identify the infinitesimal massless moduli space of the theory. We show that it
agrees with the results that have recently been obtained from a ten-dimensional
perspective where supersymmetric Minkowski solutions including the Bianchi
identity correspond to an integrable holomorphic structure, with infinitesimal
moduli calculated by its first cohomology. As has recently been noted,
interplay of complex structure and bundle deformations through holomorphic and
anomaly constraints can lead to fewer moduli than may have been expected. We
derive a relation between the number of complex structure and bundle moduli
removed from the low energy theory in this way, and give conditions for there
to be no complex structure moduli or bundle moduli remaining in the low energy
theory. The link between Yukawa couplings and obstruction theory is also
briefly discussed.
