Abstract
The geometry arising from Michelson & Strominger's study of 𝒩=4B supersymmetric quantum mechanics with superconformal D(2, 1; α)-symmetry is a hyperKähler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters α are related via changes in potentials for the HKT target spaces. For α≠0, −1, we describe how each such HKT manifold M 4m is derived from a space N 4m−4 which is quaternionic Kähler with torsion and carries an Abelian instanton.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevsky, D.V., Marchiafava, S.: Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. 171, 205–273 (1996)
Britto-Pacumio, R., Michelson, J., Strominger, A., Volovich, A.: Lectures on superconformal quantum mechanics and multi-black hole moduli spaces. In: M-theory and quantum geometry (Akureyri, 1999). NATO Sci. Ser. C Math. Phys. Sci. 556. Dordrecht: Kluwer Acad. Publ., 2000, pp. 255–283
de Wit, B., Kleijn, B., Vandoren, S.: Superconformal hypermultiplets. Nucl. Phys. B 568, 475–502 (2000) eprint arXiv:hep-th/9909228
Gibbons, G.W., Papadopoulos, G., Stelle, K.S.: HKT and OKT geometries on soliton black hole moduli spaces. Nucl. Phys. B 508(3), 623–658 (1997) eprint arXiv:hep-th/9706207
Grantcharov, G., Poon, Y.S.: Geometry of hyper-Kähler connections with torsion. Commun. Math. Phys. 213(1), 19–37 (2000) eprint arXiv:math.DG/9908015
Howe, P.S., Opfermann, A., Papadopoulos, G.: Twistor spaces for QKT manifolds. Commun. Math. Phys. 197(3), 713–727 (1998) eprint arXiv:hep-th/9710072
Ivanov, S.: Geometry of quaternionic Kähler connections with torsion. J. Geom. Phys. 41(3), 235–257 (2002)
Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2(2), 231–252 (1998) eprint arXiv:hep-th/9711200
Michelson, J., Strominger, A.: The geometry of (super) conformal quantum mechanics. Commun. Math. Phys. 213(1), 1–17 (2000) eprint arXiv:hep-th/9907191
Obata, M.: Affine connections on manifolds with almost complex, quaternion or Hermitian structure. Japan J. Math. 26, 43–79 (1956)
Pedersen, H., Poon, Y.S., Swann, A.F.: Hypercomplex structures associated to quaternionic manifolds. Diff. Geom. Appl 9, 273–292 (1998)
Poon, Y.S., Swann, A.F.: Potential functions of HKT spaces. Classical and Quantum Gravity 18(21), 4711–4714 (2001)
Swann, A.F.: HyperKähler and quaternionic Kähler geometry. Math. Ann. 289, 421–450 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by R.H. Dijkgraaf
Rights and permissions
About this article
Cite this article
Poon, Y., Swann, A. Superconformal Symmetry and HyperKähler Manifolds with Torsion. Commun. Math. Phys. 241, 177–189 (2003). https://doi.org/10.1007/s00220-003-0914-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-003-0914-5