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MiS Preprint
12/2002
Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory
Ilka Agricola
Abstract
Given a reductive homogeneous space $M=G/H$ endowed with a naturally reductive metric, we study the one-parameter family of connections $\nabla^t$ 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 $\nabla$ whose torsion $T\neq 0$ 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.
