Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory

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.