Covariant symplectic structure of the complex Monge-Ampère equation

Details Covariant symplectic structure of the complex Monge-Ampère equation Covariant symplectic structure of the complex Monge-Ampère equation

2000-04-24

arxiv.org/abs/hep-th/0004164v1

Phys.Lett. A268 (2000) 293

Physics

High Energy Physics

High Energy Physics - Theory

Physics Letters A 268 (2000) 293

Scientific paper

10.1016/S0375-9601(00)00177-8

The complex Monge-Amp\`ere equation admits covariant bi-symplectic structure for complex dimension 3, or higher. The first symplectic 2-form is obtained from a new variational formulation of complex Monge- Amp\`ere equation in the framework of the covariant Witten-Zuckerman approach to symplectic structure. We base our considerations on a reformulation of the Witten-Zuckerman theory in terms of holomorphic differential forms. The first closed and conserved Witten-Zuckerman symplectic 2-form for the complex Monge-Amp\`ere equation is obtained in arbitrary dimension and for all cases elliptic, hyperbolic and homogeneous. The connection of the complex Monge-Amp\`ere equation with Ricci-flat K\"ahler geometry suggests the use of the Hilbert action. However, we point out that Hilbert's Lagrangian is a divergence for K\"ahler metrics. Nevertheless, using the surface terms in the Hilbert Lagrangian we obtain the second Witten-Zuckerman symplectic 2-form for complex dimension>2.

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