Definition of Special Kahler manifold

Question

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

Answer 1

Special geometry is the geometry on the moduli space of (vectormultiplets) scalars in $\mathcal{N}=2$ $4d$ supergravity. A geometric realization of a $\mathcal{N}=2$ $4d$ supergravity is obtained by compactification of type IIB superstrig theory on a Calabi-Yau 3-fold $X$. In this case the moduli space is the moduli space of complex structures on $X$ and most of the objects of special geometry have a clear geometric significance. The dimension n of the moduli space is the Hodge number $h^{1,2}$ of $X$. There is clearly a natural bundle on the moduli space: the bundle whose fiber is the cohomology $H^3(X)$ of $X$, which is of rank $2n+2$, and is equipped with a symplectic form given by the intersection pairing on the cohomology group (it is symplectic because 3 is odd), there is a natural section given by the holomorphic volume form, periods of the holomorphic volume forms define the special coordinates...