What is a symplectic bundle? Is it a fibre bundle or a vector bundle? I am hoping for a not-very-technical answer because I'm not familiar with bundles in general. Sorry for that.
PS: This symplectic bundle enters in the definition of a special Kahler manifold (or in maths what they call projective Kahler manifold I suppose).
Any answer is very welcomed.
This is not terminology I've heard before, but one might predict (and it turns out to be true) that the authors here mean "symplectic vector bundle". This is a smooth vector bundle $E \to M$ such that each fiber has the (smoothly varying) structure of a symplectic vector space.
One could certainly make a notion of a fiber bundle whose fibers were symplectic manifolds but I claim this would probably not be that interesting a notion unless the base or total space had further structure (eg, Langrangian fibrations are interesting).