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Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory. (English) Zbl 1032.53041

Let \(M= G/ H\) be a reductive homogeneous space endowed with a naturally reductive metric. The author studies the one-parameter family of \(G\)-invariant connections \(\nabla_t\) joining the canonical (\(t = 0\)) and the Levi-Civita (\(t = 1/2 \)) connection. She shows that the Dirac operator \(D^t\) corresponding to \(t = 1/3\) is the “cubic Dirac operator” introduced by B. Kostant [Duke Math. J. 100, 447-501 (1999; Zbl 0952.17005)] and she gets a simple expression of the square of \(D^t\), for any \(t\), in terms of Casimir operators and scalars only, generalizing the classical Parthasarathy formula on symmetric spaces. As an application, she gets a new \(G\)-invariant first order differential operator \(\mathcal D\) on spinors that has no analogue on symmetric spaces and an eigenvalue extimate for the first eigenvalue of \(D^{1/3}\).
The above approach can be used for studying the string equations on naturally reductive spaces. Indeed, the geometric model for the common sector of type II string theories may be geometrically described by a tuple \((M, g, H, \Phi, \Psi)\) where \(H\) is a 3-form, \(\Phi\) is the so-called dilaton function and \(\Psi\) is a spinor field satisfying a coupled system of field equations. If \(\Phi\) is constant, the equations are equivalent to construct a Riemannian manifold \((M, g)\) which is Ricci flat and admits a parallel spinor with respect to some metric conne ction \(\nabla\) whose torsion \(T = H \neq 0\) is a 3-form. The number of preserved supersymmetries depends essentially on the number of \(\nabla\)-parallel spinors. The author presents some interesting results about solutions to the string equations, discussing the significance of constant spinors and showing that the string equations cannot have any solutions at all if the lifted Casimir operator is non-negative. Moreover, she gives a detailed discussion of the 5-dimensional Stiefel manifold.
Reviewer: Anna Fino (Torino)

MSC:

53C30 Differential geometry of homogeneous manifolds
53C05 Connections (general theory)
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)

Citations:

Zbl 0952.17005
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