Stability conditions for Seiberg-Witten quivers

This thesis describes a connected component of the space of numerical stability conditions of certain CY3 triangulated categories using the period map of a meromorphic differential on a family of elliptic curves. The motivation for this result comes from studying meromorphic quadratic differentials on Riemann surfaces. On the one hand, a meromorphic quadratic differential on a Riemann surface defines a double cover, its spectral curve, together with a meromorphic abelian differential on it known as the Seiberg-Witten differential. On the other hand certain strata of meromorphic quadratic differentials determine a CY3 triangulated category such that the periods of the Seiberg-Witten differential define the central charge of a stability condition on the category.

The simplest examples of this construction involve two-dimensional strata of meromorphic quadratic differentials on the Riemann sphere in which case the spectral curves are elliptic curves. There are 10 such strata in bijective correspondence with the Painlev\'{e} equations whose families of spectral elliptic curves include the original examples of Seiberg-Witten curves and certain degenerations thereof. In these cases the periods of the Seiberg-Witten differential satisfy a hypergeometric differential equation, so that its period map is described by the Schwarz triangle theorem. In all but one of these examples this period map can be lifted to a map to a canonical connected component of the space of numerical stability conditions of the associated category.