Communications in Mathematical Physics

, Volume 232, Issue 3, pp 535–563 | Cite as

Connections on Naturally Reductive Spaces, Their Dirac Operator and Homogeneous Models in String Theory

  • Ilka Agricola


 Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections ∇ 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 \(\) on spinors and an eigenvalue estimate for the first eigenvalue of D1/3. 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≠ 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 a detailed discussion of a 5-dimensional example.


Manifold String Theory Riemannian Manifold Differential Operator Symmetric Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Ilka Agricola
    • 1
  1. 1.Institut für Reine Mathematik, Humboldt-Universität zu Berlin, Sitz: WBC Adlershof, 10099 Berlin, Germany. E-mail:

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