Moduli spaces of quadratic differentials: Abel-Jacobi map and deformation

We study the moduli space of framed quadratic differentials with prescribed singularities parameterized by a decorated marked surface with punctures (DMSp), where simple zeros, double poles and higher order poles respectively correspond to decorations, punctures and boundary components. We show that the fundamental group of this space equals the kernel of the Abel-Jacobi (AJ) map from the surface mapping class group of DMSp to the first homology group of the marked surface (without decorations/punctures). Moreover, a universal cover of this space is given by the space of stability conditions on the associated 3-Calabi-Yau category. Furthermore, when we partially compactify and orbifold this moduli space by allowing the collision of simple zeros and some of the double poles, the resulting moduli space is isomorphic to a quotient of the space of stability conditions on the deformed (with respect to those collidable double poles) 3-Calabi-Yau category. Finally, we show that the fundamental group of this partially compactified orbifold equals the quotient group of the kernel of the AJ map by the square of any point-pushing diffeomorphism around any collidable double pole. This construction can produces any non-exceptional spherical/Euclidean Artin braid groups.