Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

TL;DR

This paper explores the connection between Seiberg-Witten geometries, rational elliptic surfaces, and BPS quivers in rank-one 4d N=2 theories, providing classifications, explicit functions, and insights into quiver mutations.

Contribution

It classifies modular rational elliptic surfaces related to these theories and derives explicit modular functions, linking monodromy domains to BPS quivers and their mutations.

Findings

Complete classification of modular surfaces using PSL(2,Z) subgroups

Explicit formulas for modular functions of these surfaces

Method to determine BPS quivers from monodromy domains

Abstract

We study the Coulomb branches of the rank-one 4d $\mathcal{N} = 2$ quantum field theories, including the KK theories obtained from the circle compactification of the 5d $\mathcal{N}= 1$ $E_n$ Seiberg theories. The focus is set on the relation between their Seiberg-Witten geometries and rational elliptic surfaces, with more attention being given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group ${\rm PSL}(2,\mathbb{Z})$. We derive closed-form expressions for the modular functions for all congruence and some of the non-congruence subgroups associated with these geometries. Moreover, we show how the BPS quivers of these theories can be determined directly from the fundamental domains of the monodromy groups and study how changes of these domains can be interpreted as quiver mutations. This approach can also be generalized to theories whose Coulomb branches contain `undeformable' singularities, leading to known quivers of such theories.