×

Connections on naturally reductive spaces and their Dirac operators. (Connexions sur les espaces homogènes naturellement réductifs et leurs opérateurs de Dirac.) (French) Zbl 1010.53024

In a homogeneous space whose metric is naturally reductive a set of connections is regarded, depending on a parameter \(t\). The author proves that the Dirac operator for a certain value of \(t\) coincides with the so-called cubic Dirac operator introduced by B. Kostant in 1999. Further, an eigenvalue estimate for the first eigenvalue of this operator is provided as well as some applications to string theory.
Reviewer: Johann Lang (Graz)

MSC:

53C05 Connections (general theory)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Th. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, SFB 288 preprint Nr. 492, 2001, math.DG/0102142, à paraître dans Asian J. Math
[2] Friedrich, Th., Dirac operators in Riemannian geometry, Grad. stud. math., 25, (2000), American Mathematical Society Providence, RI
[3] Gross, B.H.; Kostant, B.; Ramond, P.; Sternberg, S., The Weyl character formula, the half spin representations, and equal rank subgroups, Proc. nat. acad. sci. USA, 95, 15, 8441-8442, (1998) · Zbl 0918.17002
[4] Jensen, G., Imbeddings of Stiefel manifolds into Grassmannians, Duke math. J., 42, 3, 397-407, (1975) · Zbl 0335.53042
[5] Kobayashi, S.; Nomizu, K., Foundations of differential geometry II, Wiley classics library, (1996), Wiley Princeton · Zbl 0175.48504
[6] Kostant, B., On differential geometry and homogeneous spaces II, Proc. natl. acad. sci. USA, 42, 354-357, (1956) · Zbl 0075.31603
[7] Kostant, B., A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke math. J., 100, 3, 447-501, (1999) · Zbl 0952.17005
[8] Parthasarathy, R., Dirac operator and the discrete series, Ann. of math., 96, 1, 1-30, (1972) · Zbl 0249.22003
[9] Slebarski, S., The Dirac operator on homogeneous spaces and representations of reductive Lie groups I, Amer. J. math., 109, 283-301, (1987) · Zbl 0649.58031
[10] Strominger, A., Superstrings with torsion, Nuclear phys. B, 274, 253-284, (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.