MiS Preprint Repository

Given a reductive homogeneous space $M=G/H$ endowed with a naturally reductive metric, we study the one-parameter family of connections ${\mathrm{\nabla }}^t$ 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 $\mathcal{D}$ 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 $\mathrm{\nabla }$ whose torsion $T\ne 0$ 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.