  Given a reductive homogeneous space M=G/H endowed with a naturally reductive
metric, we study the one-parameter family of connections joining the canonical
and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D^t
corresponding to t=1/3 is the so-called ``cubic'' Dirac operator recently
introduced by B. Kostant, and derive the formula for its square for any t, thus
generalizing the classical Parthasarathy formula on symmetric spaces.
Applications include the existence of a new G-invariant first order
differential operator on spinors and an eigenvalue estimate for the first
eigenvalue of D^{1/3}. This geometric situation can be used for constructing
Riemannian manifolds which are Ricci flat and admit a parallel spinor with
respect to some metric connection whose torsion T is a 3-form, the geometric
model for the common sector of string theories. We present some results about
solutions to the string equations and give a detailed discussion of some
5-dimensional example.
