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Transactions of the American Mathematical Society

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Toric hypersymplectic quotients


Authors: Andrew Dancer and Andrew Swann
Journal: Trans. Amer. Math. Soc. 359 (2007), 1265-1284
MSC (2000): Primary 53C25; Secondary 53D20, 53C55, 57S15
DOI: https://doi.org/10.1090/S0002-9947-06-03925-0
Published electronically: August 24, 2006
MathSciNet review: 2262850
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Abstract: We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space $ \mathbb{R}^{4d}$ by the action of a compact Abelian group. These $ 4n$-dimensional quotients carry a multi-Hamilitonian action of an $ n$-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in $ \mathbb{R}^{3n}$. We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on $ \mathbb{R}^{4n}$ and metrics on complements of hypersurfaces in compact manifolds.


References [Enhancements On Off] (What's this?)

  • [AD] A. Andrada and I. Dotti, Double products and hypersymplectic structures on $ \mathbb{R}^{4n}$, Comm. Math. Phys. 262 (2006), no. 1, 1-16. MR 2200879
  • [BD] R. Bielawski and A. Dancer. The geometry and topology of toric hyperkähler manifolds. Comm. Anal. Geom. 8 (2000) 727-759. MR 1792372 (2002c:53078)
  • [De] T. Delzant. Hamiltoniens périodiques et images convexe de l'application moment. Bull. Soc. Math. France 116 (1988) 315-339. MR 0984900 (90b:58069)
  • [FPPW] A. Fino, H. Pedersen, Y. S. Poon and M. Weye Sørensen. Neutral Calabi-Yau structures on Kodaira manifolds. Commun. Math. Phys. 248 (2004), no. 2, 255-268. MR 2073135 (2005d:32044)
  • [Ga] K. Galicki. A generalisation of the momentum mapping construction for quaternionic Kähler manifolds. Commun. Math. Phys. 108 (1987) 117-138. MR 0872143 (88f:53088)
  • [GL] K. Galicki and H. B. Lawson. Quaternionic reduction and quaternionic orbifolds. Math. Annalen 282 (1988) 1-21. MR 0960830 (89m:53075)
  • [Gu] V. Guillemin. Kähler structures on toric varieties. J. Diff. Geom. 40 (1994) 285-309. MR 1293656 (95h:32029)
  • [HP] M. Harada and N. Proudfoot. Properties of the residual circle action on a hypertoric variety. Pacific J. Math. 214 (2004), no. 2, 263-284. MR 2042933 (2004k:53139)
  • [HKLR] N. J. Hitchin, A. Karlhede, U. Lindström and M. Rocek. Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108 (1987) 535-589. MR 0877637 (88g:53048)
  • [H] N. J. Hitchin. Hypersymplectic quotients. Acta Acad. Sci. Tauriensis, supplemento al numero 124 (1990) 169-180.
  • [Hu] C. M. Hull. Actions for $ (2,1)$ sigma models and strings. Nuclear Phys. B 509 (1998), no. 1-2, 252-272. MR 1601779 (98m:81144)
  • [Ka] H. Kamada. Neutral hyper-Kähler structures on primary Kodaira surfaces. Tsukuba J. Math. 23 (2) (1999) 321-332. MR 1715481 (2000g:53051)
  • [MS] L. J. Mason and G. A. J. Sparling. Nonlinear Schrödinger and Korteweg-de Vries are reductions of self-dual Yang-Mills. Phys. Lett. A 137 (1989), 29-33. MR 0995226 (90d:58169)

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Additional Information

Andrew Dancer
Affiliation: Jesus College, Oxford, OX1 3DW, United Kingdom
Email: dancer@maths.ox.ac.uk

Andrew Swann
Affiliation: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Email: swann@imada.sdu.dk

DOI: https://doi.org/10.1090/S0002-9947-06-03925-0
Keywords: Hypersymplectic structure, neutral hyperk\"ahler manifold, toric variety, moment map
Received by editor(s): September 29, 2004
Received by editor(s) in revised form: December 21, 2004
Published electronically: August 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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