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

TL;DR

This paper introduces the use of mixed Hodge structures to fully analyze the low energy physics of the Coulomb branch in four-dimensional $ abla=2$ theories, addressing longstanding challenges in extracting physical data from Seiberg-Witten geometries.

Contribution

It demonstrates how mixed Hodge structures can resolve key issues in understanding Coulomb branch solutions, including charge distinction, IR theory determination at singularities, and SW differential extraction.

Findings

Mixed Hodge structures encode electric-magnetic and flavor charges.

MHS at singular fibers determines IR physics.

MHS at infinity yields the Seiberg-Witten differential.

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.