Covariant symplectic structure of the complex Monge-Amp\`ere equation

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.Comment: Physics Letters A 268 (2000) 29

A theorem on topologically massive gravity

We show that for three dimensional space-times admitting a hypersurface orthogonal Killing vector field Deser, Jackiw and Templeton's vacuum field equations of topologically massive gravity allow only the trivial flat space-time solution. Thus spin is necessary to support topological mass.Comment: published in Classical and Quantum Gravity 13 (1996) L2

Electrovac pppp-waves

New exact solutions of the Einstein-Maxwell field equations that describe $pp$-waves are presented