Connectedness on Special Kaehler manifolds I just wanted to make a short/concise question which is quite mathematical but the aim is physical so I would like to ask it here. Anyone knows if there is a general statement about connectedness on Special Kaehler manifolds? These are of course not simply connected but maybe there is a theorem which states that they are in general path connected.
 A: I think that this question of yours is interesting and it was correctly posted in physics (not math) stack exchange. :)
I am not aware of any evidence or counterexample to the statement of connectedness of special Kahler spaces (and I suspect there is none). But allow me to say something about why some manifolds of this type may be connected in "relevant/reasonable physical situations".
1) Let $\mathscr{M}$ be the complex structure moduli space of a (non-compact) Calabi-Yau threefold $\mathscr{X}$ ( for details see chapter five of this paper) . Recall that the B-model topological string partition function is defined as a section over the Hodge bundle $\mathscr{H} \rightarrow \mathscr{M}$ over the moduli space $\mathscr{M}$ with fiber $H^{3}(\mathscr{X}) $; if it were the case that $\mathscr{M}$ was disconnected, then the B-model should develop an holomorphic anomaly (see this and this
) because now there are inequivalent choices of polarizations for $H_{3}(\mathscr{X};\mathbb{R})$ (namely, any two that belong to separate pieces of $\mathscr{M}$) i.e. different complex structures for $\mathscr{X}$. The B-model couldn't be background independent if the moduli space of complex structures for a target CY3 was disconnected.
2) Another argument for the same situation in 1) but now in the Ooguri-Strominger-Vafa regime. Recall here that $Z_{BH}(\mathscr{X}) =|Z_{Top}(\mathscr{X})|^{2}$ where $Z_{BH}(\mathscr{X})$ is the partition function a CY black hole in type IIA string theory compactified on $\mathscr{X}$ and $|Z_{Top}(\mathscr{X})|^{2}$ the squere of the topological string partition function on $\mathscr{X}$. It seems very bizarre (using the latter statistical mechanical interpretation of $Z_{Top}(\mathscr{X})$) that the black hole may recieve two separated contributions from separated components $Z_{Top}(\mathscr{X})$ (one for each disconected piece of $\mathscr{M}$), that would violate the ergodic property of the black hole "phase space" $\mathscr{M}$.
3) Possible violation of swampland criteria for effective 4d N=2 supergravity theories. I have not an argument here, but many examples of special Kahler manifolds arise as the moduli space of hypermultiplets (or vector multiplets) for theories with eight supercharges in four dimensions, it sounds bizarre (at least for the theories that belong to the landscape) that you can't be able to reach any VeV configuration for the scalars in the hypermultiplets from an arbitrary other one (possible situation if those moduli spaces were disconnected). That kind of obstruction seems to (probably) violate the "ban of non-trivial cobordism classes") from the string landscape.
I hope I have been helpful.
