

Preprint 12/2002
Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory
Ilka Agricola
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Submission date: 12. Feb. 2002
Pages: 27
published in: Communications in mathematical physics, 232 (2003) 3, p. 535-563
DOI number (of the published article): 10.1007/s00220-002-0743-y
Bibtex
MSC-Numbers: 53C27, 53C30
Keywords and phrases: kostant's dirac operator, naturally reductive space, invariant connection
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Abstract:
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
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
. 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
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.