Mixed Hodge structure and $\mathcal{N}=2$ Coulomb branch solution

Abstract: The Coulomb branch of four dimensional $\mathcal{N}=2$ theories can be solved by finding a Seiberg-Witten (SW) geometry and a SW differential. While lots of SW geometries are found, the extraction of low energy theory out of it is limited due to following reasons: (a) the difficulty of distinguishing electric-magnetic and flavor charges; (b) the difficulty of determining the low energy theory at singular point; (c) the lack of SW differential. We show that the mixed Hodge structure (MHS) can be used to fully solve the low energy physics of Coulomb branch at every vacuum. The MHS can be used to solve above three problems as follows: (a) The smooth fiber of SW geometry carries a Mixed Hodge Structure structure with two weights: one weight describes electric-magnetic charge and the other for the flavor charge; (b) for the singular fiber, there are three MHS which can be used to determine the IR theory; (c) the MHS at $\infty$ is used to find the SW differential.