Ilka Agricola
Connections on Naturally Reductive Spaces, their Dirac Operator and Homogeneous Models in String Theory
Preprint series: ESI preprints

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

53C30 Homogeneous manifolds, See also {14M15, 14M17, 32M10,

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.
Keywords: Kostant's Dirac operator, naturally reductive space, invariant connection, vanishing theorems, string equations