Spin(9)-structures and connections with totally skew-symmetric torsion. (English) Zbl 1039.53049

The problem studied in this paper originates in the type II string theory where the basic model consists of a Riemannian metric, a metric connection with totally skew-symmetric torsion, and also of a dilation function and a spinor field. The author is interested in metric connections with totally skew-symmetric torsion, establishes a criterion for existence of such a connection on a 16-dimensional Riemannian manifold with Spin(9)-structure and shows that if this criterion holds then such a connection is unique. Therewith he characterizes geometric types admitting such connections and, in particular, demonstrates that in difference with some other structures (i.e., \(G_2\)-, Spin(7)-, quaternionic Kähler and contact structures) for Spin(9)-structures the existence of a connection with totally skew-symmetric torsion is not invariant under conformal changes of a metric.


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
53C05 Connections (general theory)
Full Text: DOI arXiv


[1] I. Agricola, Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory, Commun. Math. Phys, in press. · Zbl 1032.53041
[2] I.G. Dotti, A. Fino, HyperKähler torsion structures invariant by nilpotent Lie groups. math.dg/0112166. · Zbl 1001.53031
[3] M. Fernandez, A classification of Riemannian manifolds with structure group Spin(7), Ann. de Mat. Pura ed Applicata CXLIII (1986) 101-122. · Zbl 0602.53025
[4] Friedrich, Th., Weak spin(9)-structures on 16-dimensional Riemannian manifolds, Asian J. math., 5, 129-160, (2001) · Zbl 1021.53028
[5] Th. Friedrich, On types of non-integrable geometries, Proc. Winter School Srni, in press. math.dg/0205149.
[6] Friedrich, Th.; Ivanov, S., Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. math., 6, 303-336, (2002) · Zbl 1127.53304
[7] Th. Friedrich, S. Ivanov, Almost contact manifolds and type II string equations, Crelle J., in press.
[8] Th. Friedrich, S. Ivanov, Killing spinor equations in dimension 7 and geometry of integrable G_{2}-manifolds. math.dg/0112201.
[9] W. Fulton, J. Harris, Representation Theory, Graduate Texts in Mathematics, Springer, New York, 1991. · Zbl 0744.22001
[10] Gray, A., Weak holonomy groups, Math. Z., 123, 290-300, (1971) · Zbl 0222.53043
[11] Howe, P.S.; Papadopoulos, G., Finiteness and anomalies in (4,0) supersymmetric sigma models, Nucl. phys. B, 381, 360, (1992)
[12] S. Ivanov, Connection with torsion, parallel spinors and geometry of Spin(7)-manifolds. math.dg/0111216.
[13] Ivanov, S.; Papadopoulos, G., Vanishing theorems and string background, Class. quant. grav., 18, 1089-1110, (2001) · Zbl 0990.53078
[14] S. Ivanov, I. Minchev, Quaternionic Kähler and hyperKähler manifolds with torsion and twistor spaces. math.dg/0112157.
[15] Spindel, P.; Sevrin, A.; Troost, W.; van Proeyen, A., Extended supersymmetric σ-models on group manifolds, Nucl. phys. B, 308, 662-698, (1988)
[16] Strominger, A., Superstrings with torsion, Nucl. phys. B, 274, 253-284, (1986)
[17] M. Verbitsky, HyperKähler manifolds with torsion, supersymmetry and Hodge theory. math.ag/0112215.
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.