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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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.

Received:
Feb 12, 2002
Published:
Feb 12, 2002
MSC Codes:
53C27, 53C30
Keywords:
kostant's dirac operator, naturally reductive space, invariant connection

Related publications

inJournal
2003 Repository Open Access
Ilka Agricola

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

In: Communications in mathematical physics, 232 (2003) 3, pp. 535-563