"A classical integrable system in the sense of Liouville-Arnold is a fibration X --> B, where X is symplectic and the fibres are Lagrangian tori. Locally, X = TB/L for a lattice L, and carries a torsion-free connection D. This paper studies the situation when the base B admits a symplectic structure \omega parallel with respect to D. It is shown that locally X admits a triple of symplectic structures such that the dual fibration carries a hypercomplex structure, i.e., a pair of anti-commuting integrable complex structures. This implies that X carries a hyper-Kähler metric of indefinite signature and the authors show that B carries an indefinite special Kähler metric in the sense of [D. S. Freed, Comm. Math. Phys. 203 (1999), no. 1, 31–52; MR1695113 (2000f:53060)], supplementing the results of [D. V. Alekseevskiı, V. Cortés Suárez and C. Devchand, J. Geom. Phys. 42 (2002), no. 1-2, 85–105; MR1894078 (2003i:53064)]." Andrew Swann, MR2078231 (2005g:53081).
Hyper-symplectic structures on integrable systems
BARTOCCI, CLAUDIO;
2004-01-01
Abstract
"A classical integrable system in the sense of Liouville-Arnold is a fibration X --> B, where X is symplectic and the fibres are Lagrangian tori. Locally, X = TB/L for a lattice L, and carries a torsion-free connection D. This paper studies the situation when the base B admits a symplectic structure \omega parallel with respect to D. It is shown that locally X admits a triple of symplectic structures such that the dual fibration carries a hypercomplex structure, i.e., a pair of anti-commuting integrable complex structures. This implies that X carries a hyper-Kähler metric of indefinite signature and the authors show that B carries an indefinite special Kähler metric in the sense of [D. S. Freed, Comm. Math. Phys. 203 (1999), no. 1, 31–52; MR1695113 (2000f:53060)], supplementing the results of [D. V. Alekseevskiı, V. Cortés Suárez and C. Devchand, J. Geom. Phys. 42 (2002), no. 1-2, 85–105; MR1894078 (2003i:53064)]." Andrew Swann, MR2078231 (2005g:53081).
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