A survey on hyper-Kähler with torsion geometry

• Autores: María L. Barberis
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 49, Nº. 2, 2008, págs. 121-131
• Idioma: inglés
• Enlaces
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• Resumen

  • Manifolds with special geometric structures play a prominent role in some branches of theoretical physics, such as string theory and supergravity. For instance, it is well known that supersymmetry requires target spaces to have certain special geometric properties. In many cases these requirements can be interpreted as restrictions on the holonomy group of the target space Riemannian metric. However, in some cases, they cannot be expressed in terms of the Riemannian holonomy group alone and give rise to new geometries previously unknown to mathematicians. An example of this situation is provided by hyper-Kähler with torsion (or HKT) metrics, a particular class of metrics which possess a compatible connection with torsion whose holonomy lies in Sp(n). A survey on recent results on HKT geometry is presented.

• Referencias bibliográficas
  • Banos, B., Swann, A.. (2004). Potentials for hyper-Kähler metrics with torsion. Class. Quant.Grav.. 21. 3127-3136
  • Barberis, M.L., Dotti, I., Miatello, R.. (1995). On certain locally homogeneous Clifford manifolds. Ann. Glob. Anal. Geom.. 13. 289-301
  • Barberis, M.L., Dotti, I.. (2004). Abelian complex structures on solvable Lie algebras. J. Lie Theory. 14. 25-34
  • Barberis, M.L., Dotti, I., Verbitsky, M.. Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry.
  • Barberis, M.L., Fino, A.. New strong HKT manifolds arising from quaternionic representations.
  • Benson, C., Gordon, C.S.. (1988). Kähler and symplectic structures on nilmanifolds. Topology. 27. 513-518
  • Bismut, J. M.. (1989). Local index theorem for non-Kähler manifolds. Math. Ann.. 284. 681-699
  • Cavalcanti, G.R., Gualtieri, M.. (2004). Generalized complex structures on nilmanifolds. J. Symplectic Geom.. 2. 393-410
  • Dotti, I., Fino, A.. (2002). Hyper-Kähler with torsion structures invariant by nilpotent Lie groups. Class. Quantum Grav.. 19. 1-12
  • Fino, A., Grantcharov, G.. (2004). Properties of manifolds with skew-symmetric torsion and special holonomy. Adv. Math.. 189. 439-450
  • Gates, S. J., Hull, C.M., Roek, M.. (1984). Twisted multiplets and new supersymmetric non-linear -models. Nucl. Phys. B. 248. 157-186
  • Gauduchon, P., Tod, K.P.. (1998). Hyper-Hermitian metrics with symmetry. J. Geom. Phys.. 25. 291-304
  • Gibbons, G. W., Papadopoulos, G., Stelle, K.. (1997). HKT and OKT geometries on soliton blach hole moduli space. Nucl. Phys. B. 508. 623
  • Gordon, C. S., Wilson, E. N.. (1986). The spectrum of the Laplacian on Riemannian Heisenberg manifolds. Michigan Math. J.. 33. 253-271
  • Grantcharov, G., Papadopoulos, G., Poon, Y. S.. (2002). Reduction of HKT-Structures. J. Math. Phys.. 43. 3766-3782
  • Grantcharov, G., Poon, Y. S.. (2000). Geometry of hyper-Kähler connection with torsion. Comm. Math. Phys.. 213. 19-37
  • Griffiths, P., Harris, J.. (1978). Principles of algebraic geometry. Wiley-Interscience. New York.
  • Hasegawa, K.. (1989). Minimal models of nilmanifolds. Proc. Am. Math. Soc.. 106. 65-71
  • Howe, P.S., Papadopoulos, G.. (1996). Twistor spaces for HKT manifolds. Phys. Lett. B. 379. 81-86
  • Howe, P.S., Papadopoulos, G.. (1992). Finiteness and anomalies in (4, 0) suspersymmetric sigma models for HKT manifolds. Nucl. Phys. B. 381....
  • Ivanov, S., Minchev, I.. (2004). Quaternionic Kähler and hyper-Kähler manifolds with torsion and twistor spaces. J. Reine Angew. Math.. 567....
  • Joyce, D.. (1992). Compact hypercomplex and quaternionic manifolds. J. Diff. Geom.. 35. 743-761
  • Mal'ev, A. I.. (1951). On a class of homogeneous spaces. 39.
  • Martín Cabrera, F., Swann, A.. The intrinsic torsion of almost quaternion-Hermitian manifolds.
  • Milnor, J.. (1976). Curvatures of left invariant metrics on Lie groups. Adv. Math.. 21. 293-329
  • Opfermann, A., Papadopoulos, G.. Homogeneous HKT and QKT manifolds.
  • Papadopoulos, G., Teschendorff, A.. (1998). Multi angle five brane intersections. Phys. Lett. B. 443. 159
  • Papadopoulos, G.. KT and HKT geometries in strings and in black hole moduli spaces. Proceedings of the Bonn workshop on "Special Geometric...
  • Pedersen, H., Poon, Y.S.. (1999). Inhomogeneous hypercomplex structures on homogeneous manifolds. J. Reine Angew. Math.. 516. 159-181
  • Petravchuk, P.. (1988). Lie algebras decomposable as a sum of an abelian and a nilpotent subalgebra. Ukr. Math. J.. 40. 385-388
  • Spindel, Ph., Sevrin, A., Troost, W., Van Proeyen, A.. (1988). Extended supersymmetric -models on group manifolds. Nucl. Phys. B. 308. 662-698
  • Thurston, W.P.. (1976). Some simple examples of symplectic manifolds. Proc. Am. Math. Soc.. 55. 476-478
  • Verbitsky, M.. (2002). Hyperkähler manifolds with torsion, supersymmetry and Hodge theory. Asian J. Math.. 6. 679-712
  • Verbitsky, M.. (2003). Hyperkähler manifolds with torsion obtained from hyperholomorphic bundles. Math. Res. Lett.. 10. 501-513
  • Verbitsky, M.. Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds.