I have a question on the very definition of a special Kahler manifold. One definition which is the most commonly used (and which I am most interested about) is to define this manifold by introducing a $(2n+2)$-dimensional symplectic bundle with sections $\Omega$ which are covariantly holomorphic. $$\begin{pmatrix} L^I\\ M_I \end{pmatrix}$$ See for example (String Theory and Fundamental Interactions Book edit by Maurizio Gasperini and Jnan Maharana) here.

As I am just trying to research on the subject, I find it a little too broad for me to absorb the fact of defining a manifold by introducing some bundle with some sections and starting to build up equations from them without understanding the origin of this definition. So I would be very grateful if someone could explain anything concerning the origin of this definition. Not necessarily all of it, actually any part of it will be very helpful to me.

Like why do we introduce a *symplectic* bundle and not maybe another type of bundle.

Perhaps why the sections must be covariantly holomorphic and what is meant by that precisely?

This post imported from StackExchange Physics at 2015-12-20 10:10 (UTC), posted by SE-user Beyond-formulas